All practice questions

Question 21Question →

Read the passage below and analyze the transition signals to determine the word that best completes the passage. Which word correctly fills the blank?

Fill in the blanks below

For centuries, historians characterized the medieval scholar’s reliance on authority as an uncritical submission to dogma; nevertheless, recent analytical readings of thirteenth-century scholastic disputations reveal that far from demonstrating deference, these texts engaged in rigorous dialectical interrogation, employing subtle distinctions to modify canonical texts without explicitly repudiating them.

Show answer & explanation

Answer

The blank requires a word meaning slavish, unquestioning, or excessively submissive (such as 'servile' or 'slavish'), which accurately reflects the traditional characterization of uncritical submission that the passage's contrast signals refute.

The contrast transition 'nevertheless' paired with the structural signal 'far from demonstrating {{blank_1}} deference' indicates that the blank must echo the traditional historical view of 'uncritical submission to dogma'. Therefore, an adjective indicating slavish, blind, or excessively passive submissiveness (such as 'servile' or 'slavish') is required to complete the structural logic of the sentence.

Step-by-Step Solution

1

Identify the primary macro-structural signal in the passage.

The transition word 'nevertheless' signals a sharp contrast between the traditional historical view described in the first clause and the recent analytical findings presented in the main clause.

Understanding macro-transitions establishes the overarching direction of the argument.

2

Analyze the immediate structural frame around the target blank.

The prepositional phrase 'far from demonstrating {{blank_1}} deference' sets up an internal negation. The blank describes the trait that the recent readings show was NOT actually present.

The structural clue 'far from' indicates that the missing word must echo the traditional, erroneous assumption ('uncritical submission') rather than the scholars' actual behavior.

3

Determine the required semantic meaning and tone.

To align with 'uncritical submission to dogma', the adjective modifying 'deference' must denote submissive, blind, or slavish adherence.

Matching the vocabulary tone to the baseline claim being negated ensures logical coherence across the sentence.

4

Select valid completion candidates.

Words such as 'servile', 'slavish', or 'unquestioning' fit the exact semantic profile required by the structural clues.

These terms capture the concept of blind submission rejected by the contrast phrase.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Estimated Time:1m 30s

Question 22Question →

Complete the text below by providing the word that correctly fills the blank according to the contrast pivot signal in the sentence.

Fill in the blanks below

While classical numismatists maintained that provincial minting practices during the late empire were strictly regimented, recent die-analysis demonstrates that coin production was unexpectedly , yielding substantial stylistic and compositional divergences even within single production runs.

Show answer & explanation

Answer

haphazard (or an equivalent synonym such as erratic, unmethodical, or arbitrary)

The transition word 'While' signals a contrast between the belief that minting was 'strictly regimented' and the recent evidence. The blank requires an adjective opposing strict order that aligns with 'substantial stylistic and compositional divergences', making 'haphazard' (or 'erratic' / 'unmethodical') the correct fit.

Step-by-Step Solution

1

Identify the contrast signal (pivot word) establishing sentence structure.

The opening transition 'While' establishes a contrast between the historical consensus and recent analytical findings.

Contrast pivots signal a reversal of semantic direction between clauses.

2

Analyze the clue provided in the main reference clause.

The initial clause describes minting practices as 'strictly regimented', implying high control, order, and uniformity.

The target word in the blank must express an opposing characteristic to 'strictly regimented'.

3

Deduce the word for the blank and verify against remaining context.

A word meaning lacking system or order, such as 'haphazard' or 'erratic', fits the semantic reversal and matches the elaboration 'yielding substantial stylistic and compositional divergences'.

'Haphazard' directly reverses 'strictly regimented' and creates contextual coherence.

Key Concept

Contrast and Reversal Clues

Question 23Question →

Which of the following rational numbers are strictly greater than $\frac{3}{5}$ ? Select all that apply.

Select all that apply

\frac{7}{10}

\frac{2}{3}

\frac{5}{9}

\frac{4}{7}

\frac{11}{15}

Show answer & explanation

Answer: $\frac{7}{10}$ ; $\frac{2}{3}$ ; $\frac{11}{15}$

Answer

The fractions greater than 3/5 are 7/10, 2/3, and 11/15.

Converting 3/5 to decimal form yields 0.60. Evaluating the options: 7/10 equals 0.70, 2/3 is approximately 0.667, and 11/15 is approximately 0.733. Because each of these three values is strictly greater than 0.60, all three are correct choices.

Step-by-Step Solution

1

Convert the benchmark fraction 3/5 to decimal form.

3/5 = 0.60.

Decimal conversion provides a clear standard for comparing rational numbers.

2

Convert each given choice to decimal form and compare it to 0.60.

7/10 = 0.70; 2/3 ≈ 0.667; 5/9 ≈ 0.556; 4/7 ≈ 0.571; 11/15 ≈ 0.733.

Direct comparison reveals which decimals exceed 0.60.

3

Select all fractions whose values exceed 0.60.

The values 0.70, 0.667, and 0.733 are all strictly greater than 0.60.

The corresponding fractions 7/10, 2/3, and 11/15 satisfy the given condition.

Key Concept

Comparing Rational Numbers and Fractions

Question 24Question →

Read the passage below and answer the question that follows.

While early historiographies of early modern science portrayed Isaac Newton’s esoteric laboratory notes as an embarrassing aberration that was deliberately (i)_______ by Enlightenment commentators, recent archival analyses suggest that eighteenth-century natural philosophers did not view these manuscripts with (ii)_______. Far from signaling a secret (iii)_______ of rationalist principles, Newton’s alchemical inquiries were deeply intertwined with his overarching search for a unified active principle in nature.

Which set of words, when inserted into the blanks in order, best completes the passage so that it maintains cross-sentence contextual coherence throughout?

(i) expunged, (ii) derision, (iii) repudiation

(i) lauded, (ii) reverence, (iii) confirmation

(i) expunged, (ii) reverence, (iii) repudiation

(i) publicized, (ii) derision, (iii) affirmation

(i) expunged, (ii) derision, (iii) vindication

Show answer & explanation

Answer: (i) expunged, (ii) derision, (iii) repudiation

Answer

The correct completion is the set containing 'expunged', 'derision', and 'repudiation'.

The correct set of words maintains logical, thematic, and rhetorical coherence across both sentences. 'Expunged' aligns with the portrayal of the notes as an 'embarrassing aberration', 'derision' fits the contrast showing natural philosophers did not hold the notes in contempt, and 'repudiation' establishes the concession needed before explaining that Newton's alchemical work was integrated into his broader rationalist pursuits.

Step-by-Step Solution

1

Analyze Blank (i) using the initial contrast clue 'While' and the phrase 'embarrassing aberration'.

The word in blank (i) must reflect negative treatment or removal. 'Expunged' (erased/removed) logically fits the idea that historians portrayed the notes as deliberately suppressed.

An 'embarrassing aberration' would historically be covered up or omitted rather than celebrated.

2

Analyze Blank (ii) using the shift introduced by 'recent archival analyses suggest that... did not view these manuscripts with...'.

The word in blank (ii) must represent negative judgment or contempt, which the recent analyses refute. 'Derision' (contemptuous ridicule) fits seamlessly.

Saying contemporary scholars 'did not view these manuscripts with derision' directly contrasts with the old historiographical view that they were seen as an embarrassing aberration.

3

Analyze Blank (iii) using the concessionary shift 'Far from signaling a secret... of rationalist principles, Newton’s alchemical inquiries were deeply intertwined with...'.

The word in blank (iii) must denote a rejection or abandonment of rationalism. 'Repudiation' (rejection/disavowal) completes the thought that his work was not a rejection of rationalism, but part of it.

The structure 'Far from X, [statement of Y]' requires X to represent the opposite of Y.

Key Concept

Cross-Sentence Contextual Coherence in Multi-Blank Text Completion

Question 25Question →

A baker has a flour mixture consisting only of wheat flour and rye flour. Currently, wheat flour accounts for $\frac{2}{5}$ of the total weight of the mixture. If the baker adds $9$ pounds of wheat flour to the mixture, wheat flour will account for $\frac{1}{2}$ of the new total weight of the mixture. What was the total weight, in pounds, of the original flour mixture?

Show answer & explanation

Answer: 45

Answer

The total weight of the original flour mixture was 45 pounds.

The initial total weight of the mixture is 45 pounds. Initially, wheat flour makes up

\frac{2}{5} \times 45 = 18

pounds. Adding 9 pounds of wheat flour increases the wheat flour to

18 + 9 = 27

pounds and the total weight to

45 + 9 = 54

pounds. The new fraction of wheat flour is

\frac{27}{54} = \frac{1}{2}

, which satisfies the given conditions.

Step-by-Step Solution

1

Express the initial weight of wheat flour in terms of the initial total weight

W

.

Initial weight of wheat flour =

\frac{2}{5}W

.

Wheat flour represents

\frac{2}{5}

of the total mixture.

2

Formulate an equation reflecting the addition of 9 pounds of wheat flour.

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

Adding 9 pounds of wheat flour increases both the amount of wheat flour and the total weight of the mixture by 9 pounds.

3

Solve the algebraic equation for

W

.

Cross-multiplying gives

2\left(\frac{2}{5}W + 9\right) = W + 9

, which simplifies to

\frac{4}{5}W + 18 = W + 9

. Subtracting

\frac{4}{5}W

and

9

from both sides gives

\frac{1}{5}W = 9

, so

W = 45

.

Isolating

W

gives the value of the original total weight.

Key Concept

Setting up and solving equations involving fractional parts when a quantity is added to both the part and the whole.

Question 26Question →

How many positive integers $n$ less than $30$ satisfy the condition that $n^2 - 1$ is divisible by $24$ ?

5

8

9

10

15

Show answer & explanation

Answer: 10

Answer

10

For

n^2 - 1

to be divisible by

24 = 8 \times 3

,

n

must be an odd integer that is not divisible by 3, which means

\gcd(n, 6) = 1

. The positive integers less than 30 coprime to 6 are 1, 5, 7, 11, 13, 17, 19, 23, 25, and 29, totaling 10 integers.

Step-by-Step Solution

1

Factor the expression and analyze divisibility requirements for 24.

n^2 - 1 = (n - 1)(n + 1)

. Since

24 = 8 \times 3

with

\gcd(8, 3) = 1

,

n^2 - 1

must be divisible by both

8

and

3

.

Decompose 24 into coprime prime-power factors to evaluate modular conditions independently.

2

Determine the condition for

n^2 - 1

to be divisible by 8.

If

n

is even,

n^2 - 1

is odd and cannot be divisible by 8. If

n

is odd, let

n = 2k + 1

; then

n^2 - 1 = 4k(k + 1)

. Since one of

k

or

k + 1

is always even,

4k(k + 1)

is divisible by 8. Thus,

n

must be odd.

Analyze parity requirements for the power of 2.

3

Determine the condition for

n^2 - 1

to be divisible by 3.

If

n

is a multiple of 3,

n^2 - 1 \equiv -1 \pmod 3

, which is not divisible by 3. If

n

is not a multiple of 3,

n \equiv 1

or

2 \pmod 3

, so

n^2 \equiv 1 \pmod 3

and

n^2 - 1

is divisible by 3. Thus,

n

cannot be a multiple of 3.

Analyze remainder properties modulo 3.

4

Combine the conditions and count valid positive integers

n < 30

.

The combined requirement is

\gcd(n, 6) = 1

(

n

is odd and not a multiple of 3). The positive integers less than 30 satisfying this are 1, 5, 7, 11, 13, 17, 19, 23, 25, and 29. Counting these yields 10 integers.

Enumerate all integers meeting the coprime condition within the specified domain.

Key Concept

Modular arithmetic properties of quadratic expressions and coprimality constraints

Question 27Question →

Based on the contextual clues in the sentence below, identify the vocabulary word that accurately completes the passage when used in its secondary meaning.

Fill in the blanks below

Despite the principal investigator's initial optimism regarding the clinical trial, her enthusiasm began to after several months of repetitive data collection failed to yield statistically significant results.

Show answer & explanation

Answer

The word 'flag' correctly completes the sentence, functioning in its secondary sense as a verb meaning to lose vigor, tire, or decline in intensity.

In the provided context, the transition word 'Despite' contrasts initial optimism with a subsequent decline in enthusiasm. The verb 'flag' secondary definition is to languish, tire, or lose vigor, making it the intended fit for the blank.

Step-by-Step Solution

1

Analyze the structural contrast and tone signals in the sentence.

The opening word 'Despite' signals a shift from the investigator's 'initial optimism' to a state of diminishing energy or enthusiasm caused by unpromising experimental results.

The context requires a verb that expresses a drop, weakening, or loss of momentum.

2

Evaluate high-frequency words for relevant secondary definitions.

While 'flag' is most commonly known as a noun referring to a banner or marker, its secondary verb definition specifically means to become unsteady, dwindle, or decline in vigor.

Using 'flag' in this sense provides an exact semantic match for diminishing enthusiasm.

Key Concept

Secondary and Alternative Meanings of High-Frequency Words

Question 28Question →

Let $n$ be a positive integer whose prime factorization consists only of the prime factors $2$ and $3$ . If $n$ has exactly $12$ positive divisors and $\gcd(n, 36) = 12$ , what is the value of $n$ ?

Show answer & explanation

Answer: 96

Answer

The value of

n

is

96

.

Representing

n = 2^a \times 3^b

, the number of positive divisors is

(a+1)(b+1) = 12

. The greatest common divisor

\gcd(n, 36) = \gcd(2^a \times 3^b, 2^2 \times 3^2) = 2^{\min(a,2)} \times 3^{\min(b,2)} = 12 = 2^2 \times 3^1

. This requires

\min(a,2) = 2 \implies a \ge 2

and

\min(b,2) = 1 \implies b = 1

. Substituting

b = 1

into

(a+1)(1+1) = 12

gives

2(a+1) = 12

, so

a = 5

. Therefore,

n = 2^5 \times 3^1 = 32 \times 3 = 96

.

Step-by-Step Solution

1

Set up the prime factorization of

n

and the divisor count equation.

n = 2^a \times 3^b

and

(a + 1)(b + 1) = 12

.

Since the prime factors of

n

are only

2

and

3

,

n

must take the form

2^a \times 3^b

, where the number of positive divisors is

(a+1)(b+1)

.

2

Analyze the exponent requirements using the greatest common divisor.

a \ge 2

and

b = 1

.

\gcd(2^a \times 3^b, 2^2 \times 3^2) = 2^{\min(a,2)} \times 3^{\min(b,2)} = 2^2 \times 3^1

. Matching powers gives

\min(a,2) = 2 \implies a \ge 2

, and

\min(b,2) = 1 \implies b = 1

.

3

Solve for exponent

a

and calculate

n

.

a = 5

, giving

n = 2^5 \times 3^1 = 96

.

Substituting

b = 1

into

(a+1)(1+1) = 12

gives

2(a+1) = 12 \implies a = 5

, which satisfies

a \ge 2

.

Key Concept

Prime exponent rules for GCD and divisor counting

Estimated Time:2m 0s

Question 29Question →

Let $K = 2^5 \times 3^4 \times 5^3 \times 11^2$ . How many positive integer factors of $K$ are divisible by $300$ but are not divisible by $900$ ?

16

24

36

48

96

Show answer & explanation

Answer: 24

Answer

24

The correct answer 24 is obtained by analyzing the prime factorizations

300 = 2^2 \times 3^1 \times 5^2

and

900 = 2^2 \times 3^2 \times 5^2

. Any factor

f = 2^a \times 3^b \times 5^c \times 11^d

must satisfy

a \in \{2, 3, 4, 5\}

(4 choices),

b = 1

(1 choice, since

b \ge 1

for 300 but

b < 2

for 900),

c \in \{2, 3\}

(2 choices), and

d \in \{0, 1, 2\}

(3 choices). Multiplying these choices gives

4 \times 1 \times 2 \times 3 = 24

.

Step-by-Step Solution

1

Express any factor of

K

in terms of prime factor exponent constraints.

Any factor

f

of

K

has the form

f = 2^a \times 3^b \times 5^c \times 11^d

, where

0 \le a \le 5

,

0 \le b \le 4

,

0 \le c \le 3

, and

0 \le d \le 2

.

The prime factors of

f

must be subsets of the prime factors of

K

with exponents not exceeding those in

K

.

2

Find the prime factorizations of

300

and

900

.

300 = 2^2 \times 3^1 \times 5^2

and

900 = 2^2 \times 3^2 \times 5^2

.

Divisibility criteria correspond to minimum exponent requirements for each prime factor.

3

Apply the divisibility conditions to determine constraints on each exponent.

For

f

to be divisible by

300

, we need

a \ge 2

,

b \ge 1

, and

c \ge 2

. For

f

to NOT be divisible by

900

, we must have

b < 2

. Thus,

b = 1

.

Combining

b \ge 1

and

b < 2

uniquely restricts

b

to

1

.

4

Count the number of valid choices for each exponent.

a \in \{2, 3, 4, 5\}

(

4

choices),

b \in \{1\}

(

1

choice),

c \in \{2, 3\}

(

2

choices), and

d \in \{0, 1, 2\}

(

3

choices).

The exponent

d

is unrestricted by

300

or

900

, so it can take any valid power present in

K

.

5

Multiply the number of independent choices using the fundamental counting principle.

4 \times 1 \times 2 \times 3 = 24

.

Each exponent choice can be paired independently to form a unique factor.

Key Concept

Counting Divisors Using Prime Exponent Constraints

Question 30Question →

Based on the explicit continuation and causal structural signals in the sentence below, identify the word that best completes the statement logically.

Fill in the blanks below

The extremophilic organisms discovered near the deep-sea hydrothermal vent did not merely endure the extreme thermal pressure; indeed, exposure to the superheated environment their metabolic activity, driving unprecedented rates of cellular reproduction.

Show answer & explanation

Answer

A verb conveying intensification or acceleration (such as 'accelerated', 'stimulated', or 'catalyzed')

The structural combination 'did not merely... indeed' functions as an intensifying continuation signal. The second clause must advance the narrative beyond baseline endurance. Therefore, a correct completion must express that the environmental conditions actively heightened or accelerated metabolic function.

Step-by-Step Solution

1

Analyze the structural signals in the sentence stem.

The phrase 'did not merely... indeed' establishes an intensifying continuation signal.

Recognizing structural signals reveals the logical direction and degree of momentum required for the blank.

2

Determine the contextual meaning required for the blank.

Since the organisms went beyond simple endurance, the environment must actively increase or spur their metabolic activity.

The continuation signal 'indeed' reinforces and escalates the claim from passive survival to active enhancement.

3

Formulate valid completions that fulfill the causal progression.

Words such as 'accelerated', 'stimulated', or 'catalyzed' fit the context logically.

These verbs match both the positive valence and the increased rate implied by 'driving unprecedented rates of cellular reproduction'.

Key Concept

Interpreting Continuation and Causal Structural Signals

Estimated Time:1m 0s

Question 31Question →

In circle $O$ , sector $AOB$ has a central angle of $60^\circ$ and a radius of $12$ . A smaller circle $C$ is inscribed within sector $AOB$ such that it is tangent to radii $OA$ and $OB$ , as well as to arc $AB$ . What is the area of the region inside sector $AOB$ that lies outside circle $C$ ?

4\pi

8\pi

12\pi

16\pi

18\pi

Show answer & explanation

Answer: $8\pi$

Answer

The area of the region inside sector

AOB

outside circle

C

is

8\pi

.

The correct answer is derived by first finding the area of sector

AOB

using

\frac{60}{360} \pi (12^2) = 24\pi

. Then, analyzing the geometry of the inscribed circle reveals that the line from

O

to the center of circle

C

bisects the

60^\circ

angle. In the resulting

30^\circ-60^\circ-90^\circ

right triangle, the hypotenuse length is

2r

, making the total radius of sector

AOB

equal to

2r + r = 3r = 12

, which yields

r = 4

. The area of circle

C

is

\pi (4^2) = 16\pi

. Subtracting the circle area from the sector area gives

24\pi - 16\pi = 8\pi

.

Step-by-Step Solution

1

Calculate the area of sector

AOB

Sector area

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

The area of a sector with central angle

\theta

and radius

R

is

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

.

2

Find the radius

r

of the inscribed circle

C

Distance from

O

to center of circle

C

is

OP = 2r

, so total radius

R = OP + r = 3r = 12 \implies r = 4

The line segment connecting center

O

to center

P

of circle

C

bisects the

60^\circ

angle into two

30^\circ

angles. A perpendicular dropped from

P

to radius

OA

forms a

30^\circ-60^\circ-90^\circ

right triangle where

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

, giving

OP = 2r

.

3

Calculate the area of circle

C

Area of circle

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

The area of a circle with radius

r

is

\pi r^2

.

4

Subtract the area of circle

C

from the area of sector

AOB

24\pi - 16\pi = 8\pi

The desired region is the difference between the full sector area and the enclosed circle's area.

Key Concept

Inscribed circles in sectors and central angle sector area calculations

Estimated Time:2m 30s

Question 32Question →

While early botanists viewed plant responses to pest attacks as mere passive physiological reactions, recent ecological studies indicate that flora actively emit volatile organic compounds to signal neighboring plants of impending threats. To test this phenomenon in controlled environments, researchers synthesized synthetic compounds mimicking plant distress signals and exposed healthy crops to them. Remarkably, the uninjured crops responded by preemptively synthesizing defensive enzymes. This finding strongly supports the hypothesis that plants communicate through chemical signaling, though further field trials are necessary to verify whether such interactions occur with equal efficacy in complex, unmonitored ecosystems.

Which of the following best describes the primary rhetorical function of the sentence beginning with 'Remarkably'?

It describes an empirical finding that serves as evidence supporting the hypothesis being tested.

It presents a counterargument intended to refute the early botanists' traditional perspective.

It summarizes the ultimate conclusion established by both laboratory tests and natural field observations.

It demonstrates conclusively that chemical communication functions with equal efficacy across all natural habitats.

It qualifies the author's primary hypothesis by pointing out a flaw in the researchers' initial setup.

Show answer & explanation

Answer: It describes an empirical finding that serves as evidence supporting the hypothesis being tested.

Answer

The sentence describes an empirical finding that serves as evidence supporting the hypothesis being tested.

The sentence starting with 'Remarkably' reports the observed outcome of the researchers' experiment—namely, that uninjured crops responded to distress signals by producing defensive enzymes. In the broader rhetorical structure of the passage, this observed outcome serves as empirical evidence supporting the idea that plants communicate via chemical signals.

Step-by-Step Solution

1

Analyze the context surrounding the target sentence.

The preceding sentence outlines an experimental setup (exposing healthy crops to synthetic signals). The target sentence reports what happened next (uninjured crops responded by producing enzymes).

Understanding the surrounding context clarifies how the target sentence relates to the passage's argument structure.

2

Determine the functional role of the target sentence.

The observation that uninjured crops responded demonstrates a concrete experimental result, providing empirical backing for the communication hypothesis mentioned in the subsequent sentence.

Identifying what role the sentence plays within the argument chain is key to choosing the correct description.

Key Concept

Identifying sentence function and evidence structure within an argument.

Question 33Question →

Based on the support and cause-effect clues in the passage, which words best complete the blanks to maintain sentence coherence?

Fill in the blanks below

Because the head archivist maintained an unwavering adherence to traditional preservation protocols throughout her tenure, historians deduced that the institution operated under a regime of strictly administrative practices—a systemic discipline that consequently the development of idiosyncratic filing methods.

Show answer & explanation

Answer

For the first blank, words such as 'regimented' or 'standardized' correctly align with the cause clue of 'unwavering adherence to traditional preservation protocols.' For the second blank, verbs such as 'forestalled' or 'stymied' complete the cause-effect relationship introduced by 'consequently,' reflecting how rigid standardization prevents individual variations from emerging.

The introductory clause establishes a clear cause: the archivist's 'unwavering adherence' to rules. This directly supports a descriptor for the first blank that signifies strict, uniform discipline (such as 'regimented' or 'standardized'). The phrase 'consequently' then sets up the downstream effect of this rigid environment: it prevented or hindered the emergence of individual, non-standard practices, which is logically completed by verbs like 'forestalled', 'stymied', or 'precluded'.

Step-by-Step Solution

1

Identify the cause clue driving the first blank.

The clause 'Because the head archivist maintained an unwavering adherence to traditional preservation protocols' provides direct support for the nature of the administrative practices.

The causal conjunction 'Because' dictates that the first blank must describe practices characterized by strict order, uniformity, and adherence to rules.

2

Determine the appropriate semantic fit for the first blank.

Words like 'regimented', 'standardized', or 'codified' match the required intensity of strict discipline.

These terms echo the sentiment of unwavering adherence to established rules without introducing contradictory nuance.

3

Analyze the cause-effect transition for the second blank.

The phrase 'a systemic discipline that consequently {{blank_2}}...' signals that the second blank expresses the direct outcome of a rigid system on personal variations.

A highly standardized regime naturally stops, prevents, or impedes the growth of irregular, non-standard ('idiosyncratic') practices.

4

Select the correct term for the second blank based on causal direction.

Verbs like 'forestalled', 'stymied', 'precluded', or 'stifled' fit the logical outcome.

These terms correctly capture the action of preventing or obstructing something before it can take root.

Key Concept

Support and Cause-Effect Clues in Text Completion

Question 34Question →

If $b > 1$ , which of the following expressions are equivalent to $\frac{\sqrt{b^3 \cdot \sqrt[3]{b^2}}}{b^{-\frac{1}{6}}}$ ? Select all that apply.

Select all that apply

(\sqrt[3]{b})^6

\frac{b^{\frac{7}{3}}}{b^{\frac{1}{3}}}

\sqrt{b^4 + b^2}

b^{\frac{11}{6}} \cdot b^{-\frac{1}{6}}

\left(b^{-\frac{1}{2}}\right)^{-4}

Show answer & explanation

Answer: $(\sqrt[3]{b})^6$ ; $\frac{b^{\frac{7}{3}}}{b^{\frac{1}{3}}}$ ; $\left(b^{-\frac{1}{2}}\right)^{-4}$

Answer

The equivalent expressions are

(\sqrt[3]{b})^6

,

\frac{b^{\frac{7}{3}}}{b^{\frac{1}{3}}}

, and

\left(b^{-\frac{1}{2}}\right)^{-4}

.

Simplifying the original expression yields

\frac{\sqrt{b^3 \cdot b^{2/3}}}{b^{-1/6}} = \frac{b^{11/6}}{b^{-1/6}} = b^{11/6 - (-1/6)} = b^{12/6} = b^2

. The expression

(\sqrt[3]{b})^6

equals

b^{6/3} = b^2

. The expression

\frac{b^{7/3}}{b^{1/3}}

equals

b^{7/3 - 1/3} = b^2

. The expression

\left(b^{-1/2}\right)^{-4}

equals

b^{(-1/2)(-4)} = b^2

. All three equal

b^2

.

Step-by-Step Solution

1

Simplify the expression inside the square root in the numerator.

b^3 \cdot \sqrt[3]{b^2} = b^3 \cdot b^{\frac{2}{3}} = b^{3 + \frac{2}{3}} = b^{\frac{11}{3}}

Convert radical to rational exponent and use product rule for exponents.

2

Apply the square root to the numerator.

\sqrt{b^{\frac{11}{3}}} = \left(b^{\frac{11}{3}}\right)^{\frac{1}{2}} = b^{\frac{11}{6}}

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

\frac{1}{2}

.

3

Divide by the denominator.

\frac{b^{\frac{11}{6}}}{b^{-\frac{1}{6}}} = b^{\frac{11}{6} - \left(-\frac{1}{6}\right)} = b^{\frac{12}{6}} = b^2

Apply quotient rule for exponents: subtract the exponent of the denominator from the numerator.

4

Evaluate each given option to determine which simplify to

b^2

.

The expressions

(\sqrt[3]{b})^6 = b^2

,

\frac{b^{\frac{7}{3}}}{b^{\frac{1}{3}}} = b^2

, and

\left(b^{-\frac{1}{2}}\right)^{-4} = b^2

are all equivalent.

Matching each simplified candidate expression to the target simplified value

b^2

.

Key Concept

Simplification of nested algebraic radicals and rational exponents using exponent rules

Estimated Time:2m 0s

Question 35Question →

If $x$ is a real number that satisfies the compound absolute value inequality $||2x - 5| - 7| \le 4$ , which of the following values could be the value of $x$ ? Select all such values.

Select all that apply

-2

0

3

6

9

Show answer & explanation

Answer: $-2$ ; $0$ ; $6$

Answer

The real numbers

-2

,

0

, and

6

satisfy the inequality.

The solution set to the compound inequality

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

is the union of two intervals:

[-3, 1]

and

[4, 8]

. Among the options provided,

-2

and

0

fall into the interval

[-3, 1]

, and

6

falls into the interval

[4, 8]

. Therefore, these three values satisfy the original inequality.

Step-by-Step Solution

1

Unfold the outer absolute value inequality.

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

For any expression

U

and constant

c \ge 0

,

|U| \le c

is equivalent to

-c \le U \le c

.

2

Isolate the inner absolute value term by adding

7

across all parts.

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

Adding a constant preserves inequality directions.

3

Break the compound inequality into two separate absolute value conditions.

Condition 1:

|2x - 5| \le 11

; Condition 2:

|2x - 5| \ge 3

Both conditions must hold simultaneously for

x

.

4

Solve Condition 1 (

|2x - 5| \le 11

).

-11 \le 2x - 5 \le 11 \implies -6 \le 2x \le 16 \implies -3 \le x \le 8

Expanding the bounded absolute value inequality and solving for

x

.

5

Solve Condition 2 (

|2x - 5| \ge 3

).

2x - 5 \ge 3

or

2x - 5 \le -3 \implies 2x \ge 8

or

2x \le 2 \implies x \ge 4

or

x \le 1

For

|U| \ge c

,

U \ge c

or

U \le -c

.

6

Find the intersection of the two solution sets.

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

Intersecting

[-3, 8]

with

(-\infty, 1] \cup [4, \infty)

yields

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

.

7

Test the provided choices against the solution set

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

.

-2 \in [-3, 1]

(valid),

0 \in [-3, 1]

(valid),

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

(invalid),

6 \in [4, 8]

(valid),

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

(invalid).

Values lying within the solution intervals satisfy the original inequality.

Key Concept

Solving nested absolute value inequalities using multi-step compound interval intersections.

Question 36Question →

An equilateral triangle $ABC$ with side length $6\sqrt{3}$ is inscribed in a circle with center $O$ . What is the area of sector $AOB$ , divided by $\pi$ ?

Show answer & explanation

Answer: 12

Answer

The area of sector

AOB

divided by

\pi

is 12.

Since triangle

ABC

is equilateral, its three vertices divide the circle into three congruent arcs of

120^\circ

each. Thus, central angle

\angle AOB = 120^\circ

. The relationship between the side length

s

of an inscribed equilateral triangle and the radius

R

of its circumscribed circle is

s = R\sqrt{3}

. Given

s = 6\sqrt{3}

, we solve for

R

to find

R = 6

. The area of sector

AOB

is

\frac{120^\circ}{360^\circ} \pi R^2 = \frac{1}{3} \pi (6^2) = 12\pi

. Dividing this area by

\pi

yields

12

.

Step-by-Step Solution

1

Find the central angle

\angle AOB

corresponding to side

AB

of the inscribed equilateral triangle.

\angle AOB = 120^\circ

An inscribed equilateral triangle divides the

360^\circ

circle into three equal central angles.

2

Calculate the radius

R

of the circumscribed circle from the given side length

s = 6\sqrt{3}

.

R = 6

In an inscribed equilateral triangle,

s = R\sqrt{3}

. Substituting

6\sqrt{3} = R\sqrt{3}

gives

R = 6

.

3

Compute the area of sector

AOB

and divide by

\pi

.

12

\text{Sector Area} = \frac{120^\circ}{360^\circ} \times \pi \times 6^2 = 12\pi

. Dividing by

\pi

leaves

12

.

Key Concept

Relationship between inscribed shapes, circle radii, and sector area

Question 37Question →

A right circular cylinder has base radius $r$ and height $h$ . The height of the cylinder is increased by $50\%$ , and its base radius is decreased by $20\%$ . Which of the following statements about the modified cylinder compared to the original cylinder must be true? Select all that apply.

Select all that apply

The volume of the cylinder decreases by

4\%

.

The lateral surface area of the cylinder increases by

20\%

.

The combined area of the two circular bases decreases by

20\%

.

If

h = r

, the total surface area of the cylinder decreases.

If

h = 4r

, the total surface area of the cylinder decreases.

Show answer & explanation

Answer: The volume of the cylinder decreases by $4\%$ .; The lateral surface area of the cylinder increases by $20\%$ .; If $h = r$ , the total surface area of the cylinder decreases.

Answer

The statements confirming a

4\%

volume decrease, a

20\%

lateral surface area increase, and a total surface area decrease when

h = r

are all true.

The volume of the modified cylinder decreases by

4\%

because

0.8^2 \times 1.5 = 0.96

. The lateral surface area increases by

20\%

because

0.8 \times 1.5 = 1.20

. When

h = r

, the original total surface area

4\pi r^2

reduces to

3.68\pi r^2

, confirming a decrease.

Step-by-Step Solution

1

Express original and modified dimensions algebraically.

Original: radius

= r

, height

= h

. Modified: radius

= 0.8r

, height

= 1.5h

.

Decreasing radius by

20\%

multiplies it by

1 - 0.20 = 0.80

, while increasing height by

50\%

multiplies it by

1 + 0.50 = 1.50

.

2

Calculate the ratio of modified volume to original volume.

V_{new} = \pi (0.8r)^2 (1.5h) = 0.64 \times 1.5 \pi r^2 h = 0.96 V_{old}

, indicating a

4\%

volume decrease.

Volume of a cylinder is given by

V = \pi r^2 h

.

3

Calculate the ratio of modified lateral surface area to original lateral surface area.

L_{new} = 2\pi (0.8r)(1.5h) = 2.4\pi r h = 1.20 L_{old}

, indicating a

20\%

lateral surface area increase.

Lateral surface area of a cylinder is given by

L = 2\pi r h

.

4

Evaluate the base area change and total surface area under specific height-to-radius ratios.

Base area becomes

(0.8)^2 = 0.64

of original (

36\%

decrease). For

h = r

, total surface area changes from

4\pi r^2

to

3.68\pi r^2

(decrease). For

h = 4r

, total surface area changes from

10\pi r^2

to

10.88\pi r^2

(increase).

Total surface area is the sum of lateral surface area and base area,

A = 2\pi r^2 + 2\pi r h

.

Key Concept

Impact of dimensional scaling on volume, lateral surface area, and total surface area of cylinders

Question 38Question →

A quality control engineer records the thickness, in millimeters, of 9 sample components: $11, 13, 15, 17, 19, 21, 23, 25,$ and $27$ . Each thickness measurement $x$ is then converted to a scaled rating $y$ using the linear formula $y = 1.5x + 4.8$ . What is the interquartile range (IQR) of the transformed dataset of $y$ -values?

Show answer & explanation

Answer: 15

Answer

The interquartile range of the transformed dataset is 15.

The interquartile range (IQR) measures the spread of the middle 50% of the data (

Q_3 - Q_1

). For the original dataset

11, 13, 15, 17, 19, 21, 23, 25, 27

, the median is

19

.

Q_1

is the median of the lower half

\{11, 13, 15, 17\}

, which is

\frac{13+15}{2} = 14

.

Q_3

is the median of the upper half

\{21, 23, 25, 27\}

, which is

\frac{23+25}{2} = 24

. Thus, the original

\text{IQR} = 24 - 14 = 10

. Under a linear transformation

y = ax + b

, measures of position shift by

a x + b

, so

Q_1(y) = 1.5(14) + 4.8 = 25.8

and

Q_3(y) = 1.5(24) + 4.8 = 40.8

. Subtracting these yields

\text{IQR}(y) = 40.8 - 25.8 = 15

. Notice that this is simply

1.5 \times 10

, as adding a constant shifts the location of the distribution but leaves measures of spread unchanged.

Step-by-Step Solution

1

Find the quartiles of the original 9-element dataset.

Q_1 = 14

and

Q_3 = 24

The median of the dataset is 19 (the 5th value). The lower half of the data consists of

11, 13, 15, 17

, so

Q_1 = \frac{13 + 15}{2} = 14

. The upper half consists of

21, 23, 25, 27

, so

Q_3 = \frac{23 + 25}{2} = 24

.

2

Calculate the interquartile range of the original dataset.

\text{IQR}_x = 10

\text{IQR}_x = Q_3 - Q_1 = 24 - 14 = 10

.

3

Apply the linear transformation rules to find the transformed interquartile range.

\text{IQR}_y = 15

For a linear transformation

y = ax + b

, the interquartile range scales by

|a|

, so

\text{IQR}_y = |a| \cdot \text{IQR}_x = 1.5 \times 10 = 15

. The constant addition of

4.8

shifts all values equally and does not affect the spread/IQR.

Key Concept

Effect of linear transformations on measures of dispersion (IQR, standard deviation, range)

Alternative Method

Transform the individual quartiles directly:

Q_1(y) = 1.5(14) + 4.8 = 25.8

and

Q_3(y) = 1.5(24) + 4.8 = 40.8

. Then calculate the new IQR directly as

40.8 - 25.8 = 15

.

Estimated Time:1m 30s

Question 39Question →

A commercial coffee roaster creates a custom blend by mixing two existing bean blends: Blend X and Blend Y. Blend X consists of 60% Arabica beans and 40% Robusta beans by weight, whereas Blend Y consists of 30% Arabica beans and 70% Robusta beans by weight. The roaster mixes a quantity of Blend X with a quantity of Blend Y to produce a total of 50 pounds of a new mixture that is 42% Arabica beans by weight. How many pounds of Blend X are in the final mixture?

Show answer & explanation

Answer: 20

Answer

20 pounds

Let

x

represent the number of pounds of Blend X. The remaining weight of the mixture,

(50 - x)

pounds, comes from Blend Y. Setting up the equation for the total weight of Arabica beans gives

0.60x + 0.30(50 - x) = 0.42(50)

. Simplifying this expression yields

0.60x + 15 - 0.30x = 21

, which reduces to

0.30x = 6

. Dividing by

0.30

gives

x = 20

. Therefore, 20 pounds of Blend X were used.

Step-by-Step Solution

1

Define variables for component weights

Let

x

be the pounds of Blend X. The weight of Blend Y used is

50 - x

pounds.

The total combined weight of the mixture is given as 50 pounds.

2

Formulate an equation for the total weight of Arabica beans

0.60x + 0.30(50 - x) = 0.42(50)

, which simplifies to

0.60x + 15 - 0.30x = 21

.

The sum of Arabica beans contributed by each blend must equal the total weight of Arabica beans in the combined mixture.

3

Solve the linear equation for

x

0.30x + 15 = 21 \implies 0.30x = 6 \implies x = 20

.

Subtract 15 from both sides to isolate the variable term, then divide by 0.30.

Key Concept

Linear Modeling and Mixture Problems

Estimated Time:1m 30s

Question 40Question →

If $x$ is a real number that satisfies the inequality $|5 - 2x| \le 7$ , what is the minimum possible value of the expression $3 - 4x$ ?

-21

-1

7

27

-27

Show answer & explanation

Answer: $-21$

Answer

-21

Solving

|5 - 2x| \le 7

yields

-7 \le 5 - 2x \le 7

. Subtracting

5

gives

-12 \le -2x \le 2

. Dividing by

-2

and flipping the inequality direction gives

-1 \le x \le 6

. To minimize

3 - 4x

, we choose the maximum value of

x

because the term

-4x

decreases as

x

increases. Substituting

x = 6

yields

3 - 4(6) = -21

.

Step-by-Step Solution

1

Unpack the absolute value inequality into a compound inequality

-7 \le 5 - 2x \le 7

For any real constant

k \ge 0

,

|u| \le k

is equivalent to

-k \le u \le k

.

2

Isolate the variable term by subtracting

5

from all parts of the inequality

-12 \le -2x \le 2

Subtracting a constant from all parts preserves inequality direction.

3

Divide all parts by

-2

and reverse the inequality signs

6 \ge x \ge -1

, which is equivalent to

-1 \le x \le 6

Dividing an inequality by a negative number reverses the direction of the inequality signs.

4

Find the minimum value of

3 - 4x

over the interval

[-1, 6]

The minimum value occurs at

x = 6

:

3 - 4(6) = 3 - 24 = -21

Because the linear expression

3 - 4x

has a negative coefficient for

x

, it is a decreasing function; its minimum occurs at the largest allowed value of

x

.

Key Concept

Linear Inequalities and Absolute Value Bounds

Estimated Time:1m 30s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution