All practice questions

Question 1921Question →

Although the committee chairman was initially revered for his uncompromising principles, subsequent archival discoveries revealed that his public stance had been largely ______ , calculated more to cultivate a pious reputation than to express genuine moral conviction.

Which TWO of the following words, when inserted into the blank, produce sentences that are equivalent in meaning?

Select all that apply

sanctimonious

austere

spartan

hypocritical

disingenuous

pious

Show answer & explanation

Answer: sanctimonious; hypocritical

Answer

The words 'sanctimonious' and 'hypocritical' complete the sentence by creating logically coherent and equivalent sentences.

The sentence contrasts the chairman's public reputation for uncompromising principles with the reality that his stance was motivated by a desire to appear devout without holding genuine conviction. The terms 'sanctimonious' and 'hypocritical' both describe a false or affected display of virtue and moral superiority, ensuring that both selections result in sentences with identical meaning.

Step-by-Step Solution

1

Analyze structural pivots and sentence context

The opening transition 'Although' signals a shift between the chairman's perceived 'uncompromising principles' and the reality. The clause 'calculated more to cultivate a pious reputation than to express genuine moral conviction' indicates that the blank must describe false, insincere, or affected moral uprightness.

Identifying the target tone and meaning required to complete the contrast setup.

2

Evaluate candidate words for contextual fit

'Sanctimonious' (making a show of being morally superior or pious) and 'hypocritical' (insincerely professing virtues one does not possess) both accurately fill the blank.

Both words match the requirement of insincere moral pretense.

3

Verify pair equivalence and eliminate distractors

'Sanctimonious' and 'hypocritical' form a valid near-synonym pair in this context. While 'austere' and 'spartan' form a synonym pair, they mean strict/severe and fail contextual fit. 'Disingenuous' fits context individually but lacks a synonym partner. 'Pious' creates a logical contradiction.

Sentence Equivalence requires selecting two words that produce identical overall sentence meanings.

Key Concept

Identifying Synonym and Near-Synonym Pairs in Context

Question 1922Question →

An agricultural cooperative harvested crops from three distinct farming zones: Zone A, Zone B, and Zone C. In Season 1, Zone A accounted for $40\%$ of the cooperative's total crop yield, Zone B accounted for $35\%$ , and Zone C accounted for the remaining $25\%$ .

In Season 2:
- The crop yield in Zone A increased by $15\%$ compared to Season 1.
- The crop yield in Zone B decreased by $20\%$ compared to Season 1.
- The crop yield in Zone C increased by $p\%$ compared to Season 1.

If the cooperative's total crop yield in Season 2 was $6\%$ greater than its total crop yield in Season 1, what is the value of $p$ ?

7

11

24

28

32

Show answer & explanation

Answer: 28

Answer

The value of

p

is

28

.

The correct value of

p

is

28

. Calculating the weighted percentage contributions of each zone shows that Zone A contributes a yield equal to

46\%

of the original total yield and Zone B contributes

28\%

, making their combined Season 2 contribution

74\%

of the original total yield. To reach the required total Season 2 yield of

106\%

of the original total, Zone C must contribute

106\% - 74\% = 32\%

of the original total yield. Since Zone C initially produced

25\%

of the total yield, the percentage increase for Zone C is

\frac{32 - 25}{25} \times 100\% = \frac{7}{25} \times 100\% = 28\%

.

Step-by-Step Solution

1

Define total initial yield and express Season 1 yields for each zone in terms of total yield

Y

.

Zone A yield =

0.40Y

, Zone B yield =

0.35Y

, Zone C yield =

0.25Y

.

Establishing baseline values based on given initial percentage shares.

2

Calculate the Season 2 yields for Zones A and B after their respective percentage changes.

Zone A yield in Season 2 =

0.40Y \times 1.15 = 0.46Y

. Zone B yield in Season 2 =

0.35Y \times 0.80 = 0.28Y

.

Applying a 15% increase to Zone A and a 20% decrease to Zone B.

3

Set up the total Season 2 yield equation using the given 6% overall increase.

0.46Y + 0.28Y + 0.25Y\left(1 + \frac{p}{100}\right) = 1.06Y

.

The sum of all three zone yields in Season 2 equals 106% of the total Season 1 yield.

4

Solve for

p

.

0.74 + 0.25\left(1 + \frac{p}{100}\right) = 1.06 \implies 0.25\left(1 + \frac{p}{100}\right) = 0.32 \implies 1 + \frac{p}{100} = 1.28 \implies p = 28

.

Subtracting 0.74 from 1.06 gives 0.32, and dividing 0.32 by 0.25 yields 1.28, representing a 28% increase.

Key Concept

Weighted Percent Change and Base Values

Question 1923Question →

A craft brewery blends two batches of cider. Batch A contains a $12\%$ sugar solution by volume, and Batch B contains a $20\%$ sugar solution by volume. The brewer mixes $15$ liters of Batch A with $x$ liters of Batch B to create a resulting mixture that is $17\%$ sugar by volume. What is the value of $x$ ?

Show answer & explanation

Answer: 25

Answer

The value of

x

is

25

.

Equating the total amount of pure sugar before and after mixing gives

0.12(15) + 0.20x = 0.17(15 + x)

. Simplifying yields

1.8 + 0.20x = 2.55 + 0.17x

, which reduces to

0.03x = 0.75

, giving

x = 25

.

Step-by-Step Solution

1

Calculate the volume of pure sugar contributed by Batch A

0.12 \times 15 = 1.8

liters of pure sugar

Batch A is

12\%

sugar by volume and has a total volume of

15

liters.

2

Express the total volume of pure sugar in terms of

x

Total sugar volume =

1.8 + 0.20x

liters

Batch B adds

x

liters of a

20\%

sugar solution.

3

Set up the mixture equation

1.8 + 0.20x = 0.17(15 + x)

The final mixture has a total volume of

(15 + x)

liters with a concentration of

17\%

sugar.

4

Expand and solve the linear equation for

x

1.8 + 0.20x = 2.55 + 0.17x \implies 0.03x = 0.75 \implies x = 25

Subtract

0.17x

and

1.8

from both sides, then divide by

0.03

.

Key Concept

Linear Equations in One Variable (Mixture Word Problems)

Estimated Time:1m 30s

Question 1924Question →

For all real numbers $x$ such that $x \neq 5$ , which of the following expressions is equivalent to $\frac{x^2 - 25}{3x - 15}$ ?

\frac{x - 5}{3}

x + 5

\frac{x + 5}{3}

\frac{x - 25}{3}

\frac{x + 5}{15}

Show answer & explanation

Answer: $\frac{x + 5}{3}$

Answer

The expression

\frac{x + 5}{3}

is equivalent to the given rational expression for all

x \neq 5

.

Factoring the numerator as a difference of squares gives

(x - 5)(x + 5)

, and factoring the denominator gives

3(x - 5)

. Dividing out the common factor

(x - 5)

simplifies the expression to

\frac{x + 5}{3}

.

Step-by-Step Solution

1

Factor the numerator using the difference of squares identity.

x^2 - 25 = (x - 5)(x + 5)

The difference of squares

a^2 - b^2

factors into

(a - b)(a + b)

.

2

Factor out the greatest common factor from the denominator.

3x - 15 = 3(x - 5)

Both terms in

3x - 15

share a common factor of

3

.

3

Divide out the common binomial factor

(x - 5)

from the numerator and denominator.

\frac{(x - 5)(x + 5)}{3(x - 5)} = \frac{x + 5}{3}

Since

x \neq 5

, the factor

(x - 5)

is non-zero and can be cancelled.

Key Concept

Simplifying rational algebraic expressions by factoring difference of squares and common linear factors.

Question 1925Question →

An industrial laboratory prepares a batch of Alloy $Z$ weighing $260$ pounds, which consists solely of copper and zinc in a ratio of $7 : 6$ by weight. A technician then adds $40$ pounds of pure copper to this batch to create Alloy $W$ . What is the ratio of copper to zinc in Alloy $W$ ?

3 : 2

3 : 5

9 : 8

7 : 6

2 : 3

Show answer & explanation

Answer: 3 : 2

Answer

3 : 2

In Alloy Z, the total weight of 260 pounds is divided into 13 equal parts of 20 pounds each. Thus, copper accounts for 7 parts (140 pounds) and zinc accounts for 6 parts (120 pounds). Adding 40 pounds of pure copper increases the copper content to 180 pounds, while zinc remains at 120 pounds. Comparing copper to zinc yields 180 : 120, which simplifies to 3 : 2.

Step-by-Step Solution

1

Determine total ratio parts for Alloy Z.

The total number of ratio parts is

7 + 6 = 13

parts.

To find the weight of each component from a part-to-part ratio, divide the total weight by total ratio parts.

2

Calculate initial weights of copper and zinc in Alloy Z.

Copper weight

= 260 \times \frac{7}{13} = 140

pounds; Zinc weight

= 260 \times \frac{6}{13} = 120

pounds.

Multiply total weight by each component's fraction of the total.

3

Update component weights after adding pure copper.

New copper weight

= 140 + 40 = 180

pounds; Zinc weight remains

120

pounds.

Only copper is added to the mixture, leaving zinc weight unchanged.

4

Compute and simplify the final ratio of copper to zinc for Alloy W.

Ratio of copper to zinc

= 180 : 120 = 3 : 2

.

Divide both terms by their greatest common divisor, 60.

Key Concept

Ratios and Component Mixture Adjustment

Question 1926Question →

Read the following passage regarding early modern scientific methodology:

In his reevaluation of early modern scientific methodology, historian Julian Vance does not outright reject the traditional narrative that empirical observation abruptly supplanted scholasticism; rather, he seeks to qualify it. By emphasizing that late medieval scholars frequently engaged in systematic observational practices long before the Scientific Revolution, Vance demonstrates that the transition was incremental rather than revolutionary, thereby refining overly simplistic historical assertions.

In the context of the passage, which of the following best captures the meaning of the word "qualify"?

limit or moderate the scope of

demonstrate competence or eligibility for

categorically refute the validity of

provide empirical verification for

endorsingly elaborate upon without altering

Show answer & explanation

Answer: limit or moderate the scope of

Answer

In the context of the passage, the word 'qualify' means to limit or moderate the scope of a statement or claim.

The sentence employs a negative-contrast structure: 'does not outright reject... rather, he seeks to qualify it.' The signal 'rather' alongside the clarification that Vance refines 'overly simplistic assertions' demonstrates that 'qualify' carries its secondary academic definition of modifying, limiting, or moderating a claim.

Step-by-Step Solution

1

Analyze the structural pivot marker

Identified the contrast construction: 'does not outright reject [X]; rather, he seeks to [target word] it.'

The transition signal 'rather' indicates a shift to an action that lies between total rejection and complete acceptance.

2

Examine the elaboration in the subsequent sentence

The text explains that Vance shows the shift was 'incremental rather than revolutionary' to refine 'overly simplistic historical assertions.'

This elaboration demonstrates that Vance is adjusting and restricting the boundaries of the traditional claim rather than abandoning it or accepting it as originally stated.

3

Match contextual meaning with option choices

Selected the option meaning 'limit or moderate the scope of'.

Secondary definitions of 'qualify' include to modify, limit, or restrict a statement, perfectly aligning with the structural context.

Key Concept

Deciphering secondary word meanings using contrast transition signals and clause relationships

Question 1927Question →

Read the passage below and complete the sentence by providing the word that best fits the blank based on the structural clues and contrast signals.

Fill in the blanks below

In nineteenth-century historiography of the Ancient Near East, early scholars categorized Bronze Age administrative tablets as strictly matter-of-fact record-keeping tools. Far from being merely ledgers of grain distribution and livestock inventories, however, recent epigraphic decipherments demonstrate that these inscriptions were crafted with subtle rhetorical flourishes designed to reinforce royal lineage and divine legitimacy.

Show answer & explanation

Answer

The word that best completes the blank is 'prosaic' (or synonymous terms such as 'mundane', 'pedestrian', or 'utilitarian').

The correct response requires recognizing the structural contrast signaled by 'Far from being merely {{blank_1}}... however'. The first sentence characterizes early scholarship's view of the tablets as 'strictly matter-of-fact record-keeping tools.' The second sentence introduces new findings showing the tablets contained 'subtle rhetorical flourishes' promoting 'divine legitimacy.' The missing word must therefore describe ledgers that are commonplace, unadorned, or purely functional—making words like 'prosaic', 'mundane', 'pedestrian', or 'utilitarian' contextually precise.

Step-by-Step Solution

1

Identify the primary context and initial descriptor.

The initial sentence establishes that early scholars viewed the administrative tablets as 'strictly matter-of-fact record-keeping tools.'

This establishes the initial baseline concept against which the rest of the sentence will react.

2

Analyze the structural signals and transition markers.

The clause begins with the concessive/contrast signal 'Far from being merely...' and is reinforced by the contrast pivot marker 'however'.

These signals signal a sharp opposition between the traditional expectation of the ledgers and the actual discovery of 'subtle rhetorical flourishes' and 'divine legitimacy.'

3

Deduce the required tone and definition for the blank.

The word filling the blank must align with 'matter-of-fact' and contrast with 'rhetorical flourishes.' It must convey that the ledgers were ordinary, uninspired, or purely functional.

Selecting 'prosaic', 'mundane', or 'utilitarian' completes the contrast created by the structural pivot.

Key Concept

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Estimated Time:1m 15s

Question 1928Question →

Read the passage below. Determine which terms best complete each corresponding blank to maintain cross-sentence contextual coherence.

Fill in the blanks below

For decades, ethnoarchaeologists believed that ancient low-temperature pottery firing was entirely , assuming that early potters lacked precise temperature control mechanisms. However, recent thermal analysis of ceramic vitrification reveals that these artisans utilized subtle draft adjustments to achieve astonishingly internal temperatures across separate firings. Consequently, scholars have begun to earlier characterizations of prehistoric pyrotechnology as rudimentary.

Show answer & explanation

Answer

Blank (i): unpredictable; Blank (ii): consistent; Blank (iii): reconsider

The initial sentence establishes an assumption of primitive, uncontrolled technology, making 'unpredictable' the appropriate choice for the first blank. The transition word 'However' introduces contradictory empirical evidence demonstrating precise thermal regulation, requiring 'consistent' for the second blank. Finally, 'Consequently' signals a logical shift in academic consensus resulting from this finding, necessitating 'reconsider' for the third blank.

Step-by-Step Solution

1

Analyze the structural and semantic clues in the first sentence.

The clause 'assuming that early potters lacked precise temperature control mechanisms' establishes that scholars viewed the firing outcomes as erratic, supporting 'unpredictable'.

The blank must reflect an outcome resulting from a lack of control mechanisms.

2

Identify the contrast signal introducing the second sentence.

The discourse marker 'However' indicates a pivot away from the traditional view, showing that 'subtle draft adjustments' produced stable heat levels, requiring 'consistent'.

The finding refutes the assumption of unpredictability by demonstrating thermal control.

3

Evaluate the logical consequence indicated in the final sentence.

The transition 'Consequently' signals that the newly demonstrated technological skill forces researchers to reassess old beliefs, necessitating 'reconsider'.

Evidence of sophisticated craft control directly undermines earlier claims that the technology was rudimentary.

Key Concept

Cross-Sentence Contextual Coherence in Multi-Blank Text Completion

Estimated Time:1m 30s

Question 1929Question →

Let $n = 2^a \cdot 3^b \cdot p^c$ , where $a$ , $b$ , and $c$ are positive integers and $p$ is a prime number strictly greater than $5$ . The integer $n$ has exactly $36$ positive divisors, and $\gcd(n, 180) = 36$ . Which of the following statements MUST be true? Select all such statements.

Select all that apply

The integer

n

is divisible by

36

.

The exponent

c

cannot exceed

3

.

The sum of the exponents

a + b + c

must equal

7

.

The least common multiple of

n

and

180

is equal to

5n

.

The exponent

a

must equal

2

.

Show answer & explanation

Answer: The integer $n$ is divisible by $36$ .; The exponent $c$ cannot exceed $3$ .; The least common multiple of $n$ and $180$ is equal to $5n$ .

Answer

The statements asserting that

n

is divisible by

36

, that the exponent

c

cannot exceed

3

, and that

\text{lcm}(n, 180) = 5n

must be true.

The statement that

n

is divisible by

36

is true because

a \ge 2

and

b \ge 2

. The statement that

c \le 3

is true because

(a+1)(b+1) \ge 9

, bounding

c+1 \le 4

. The statement that

\text{lcm}(n, 180) = 5n

is true because

\max(a, 2) = a

,

\max(b, 2) = b

, and

180

contributes a factor of

5^1

.

Step-by-Step Solution

1

Analyze the GCD condition to establish bounds on exponents

a

and

b

.

a \ge 2

and

b \ge 2

.

Since

180 = 2^2 \cdot 3^2 \cdot 5^1

and

\gcd(n, 180) = 36 = 2^2 \cdot 3^2

, the minimum of the exponents of

2

is

2

(so

a \ge 2

) and the minimum of the exponents of

3

is

2

(so

b \ge 2

).

2

Use the divisor count formula

(a+1)(b+1)(c+1) = 36

to find constraints on

c

.

c \le 3

.

Since

a \ge 2 \implies a+1 \ge 3

and

b \ge 2 \implies b+1 \ge 3

, the product

(a+1)(b+1) \ge 9

. Therefore,

c+1 = \frac{36}{(a+1)(b+1)} \le \frac{36}{9} = 4

, which gives

c \le 3

.

3

Evaluate the divisibility of

n

by

36

.

n

is divisible by

36

.

Because

a \ge 2

and

b \ge 2

,

2^2 \cdot 3^2 = 36

divides

2^a \cdot 3^b \cdot p^c = n

.

4

Calculate

\text{lcm}(n, 180)

.

\text{lcm}(n, 180) = 5n

.

\text{lcm}(2^a \cdot 3^b \cdot p^c, 2^2 \cdot 3^2 \cdot 5^1) = 2^{\max(a,2)} \cdot 3^{\max(b,2)} \cdot 5^1 \cdot p^c = 2^a \cdot 3^b \cdot 5 \cdot p^c = 5n

since

a \ge 2

,

b \ge 2

, and

p > 5

.

5

Test counterexamples for statement constancy regarding

a+b+c=7

and

a=2

.

Neither

a+b+c=7

nor

a=2

is required.

The choice

(a, b, c) = (5, 2, 1)

satisfies

(5+1)(2+1)(1+1) = 36

and

\gcd(n, 180) = 36

, yielding

a+b+c=8 \neq 7

and

a=5 \neq 2

.

Key Concept

Prime Factorization, Greatest Common Divisor (GCD), and Least Common Multiple (LCM) properties

Question 1930Question →

Rather than viewing the archivist's uncatalogued personal diaries as (i)________ reflections of trivial personal matters, recent historians contend that these candid entries provide an indispensable window into the era's broader socio-political currents; nevertheless, the diaries' fragmented nature requires scholars to exercise considerable (ii)________ when drawing sweeping structural conclusions.

Which pair of word choices best completes the passage?

(i) parochial, (ii) circumspection

(i) profound, (ii) circumspection

(i) parochial, (ii) temerity

(i) tendentious, (ii) apathy

(i) profound, (ii) temerity

Show answer & explanation

Answer: (i) parochial, (ii) circumspection

Answer

The pair '(i) parochial, (ii) circumspection' logically completes the passage.

The correct option pairs 'parochial' with 'circumspection'. The initial contrast signal 'Rather than viewing...' sets up an opposition between the traditional view of the diaries as narrow or limited ('parochial') and the modern view of them as illuminating broad historical trends. The secondary pivot 'nevertheless' introduces a qualification: because the diaries are 'fragmented', historians must exercise prudence and caution ('circumspection') when drawing broad conclusions.

Step-by-Step Solution

1

Analyze the structural pivot for Blank (i)

The opening phrase 'Rather than viewing... as (i)______ reflections of trivial personal matters, recent historians contend that... broader socio-political currents' creates a direct contrast between the old perception (narrow/small-scale) and the new historical perspective (broad/significant).

Blank (i) must mean narrow, limited, or provincial to mirror 'trivial personal matters' and contrast with 'broader socio-political currents'.

2

Evaluate candidates for Blank (i)

'Parochial' means limited or narrow in scope, which fits the contrast perfectly. 'Profound' is the opposite of trivial, breaking the contrast. 'Tendentious' means biased, which is not supported by the context of triviality.

Only 'parochial' establishes the required semantic valence for Blank (i).

3

Analyze the structural pivot for Blank (ii) based on Blank (i) and passage context

The semicolon followed by 'nevertheless' signals a concession. Even though the diaries offer valuable broad insights, their 'fragmented nature' demands that scholars proceed with care.

Blank (ii) requires a noun meaning prudence, care, or caution when forming sweeping conclusions from incomplete data.

4

Evaluate candidates for Blank (ii) and verify joint alignment

'Circumspection' means caution and prudence, perfectly matching the warning about 'fragmented nature'. 'Temerity' (foolhardy boldness) and 'apathy' (lack of interest) fail contextually.

Combining 'parochial' for Blank (i) and 'circumspection' for Blank (ii) satisfies both local syntax and inter-blank logical dependency.

Key Concept

Multi-Blank Dependency Tracking

Estimated Time:1m 30s

Question 1931Question →

On the real number line, points $P$ , $Q$ , and $R$ have coordinates $x$ , $y$ , and $z$ , respectively, such that $|x - 2| = 5$ , $|y + 4| = 3$ , and $z$ is the midpoint of segment $PQ$ . If $x < y$ , what is the value of $z$ ?

Show answer & explanation

Answer: -2

Answer

The coordinate of point

R

(the value of

z

) is

-2

.

Solving the two absolute value equations yields

x \in \{-3, 7\}

and

y \in \{-7, -1\}

. Testing the given constraint

x < y

across all four possible ordered pairs reveals that only

x = -3

and

y = -1

satisfy the condition, since

-3 < -1

is true while all other pairs fail. The midpoint of points with coordinates

-3

and

-1

is

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

.

Step-by-Step Solution

1

Solve the absolute value equation

|x - 2| = 5

.

x = 7

or

x = -3

The equation

|x - 2| = 5

splits into

x - 2 = 5 \implies x = 7

and

x - 2 = -5 \implies x = -3

.

2

Solve the absolute value equation

|y + 4| = 3

.

y = -1

or

y = -7

The equation

|y + 4| = 3

splits into

y + 4 = 3 \implies y = -1

and

y + 4 = -3 \implies y = -7

.

3

Evaluate all candidate pairs

(x, y)

under the inequality constraint

x < y

.

The only valid pair is

x = -3

and

y = -1

.

Comparing all four combinations:

7 < -1

(false),

7 < -7

(false),

-3 < -7

(false), and

-3 < -1

(true).

4

Calculate the midpoint

z

of the segment

PQ

.

z = -2

The midpoint of coordinates

-3

and

-1

on a number line is given by their average:

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

.

Key Concept

Absolute Value Equations and Midpoint on the Real Number Line

Question 1932Question →

If $3^x + 3^{-x} = 4$ , what is the value of $\frac{27^x + 27^{-x} - 2}{9^x + 9^{-x} + 1}$ ?

\frac{10}{3}

\frac{50}{17}

\frac{62}{17}

\frac{74}{15}

\frac{50}{19}

Show answer & explanation

Answer: $\frac{10}{3}$

Answer

\frac{10}{3}

Squaring

3^x + 3^{-x} = 4

yields

9^x + 2 + 9^{-x} = 16

, which simplifies to

9^x + 9^{-x} = 14

. Utilizing the sum of cubes identity gives

27^x + 27^{-x} = (3^x + 3^{-x})(9^x - 1 + 9^{-x}) = 4(14 - 1) = 52

. Substituting these values into the given fraction gives

\frac{52 - 2}{14 + 1} = \frac{50}{15} = \frac{10}{3}

.

Step-by-Step Solution

1

Square the given expression

3^x + 3^{-x} = 4

to determine

9^x + 9^{-x}

.

(3^x + 3^{-x})^2 = 9^x + 2(3^x)(3^{-x}) + 9^{-x} = 9^x + 2 + 9^{-x} = 16

, which yields

9^x + 9^{-x} = 14

.

Expanding the square of a binomial requires accounting for the middle term

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

.

2

Express

27^x + 27^{-x}

using the sum of cubes factorization identity.

27^x + 27^{-x} = (3^x)^3 + (3^{-x})^3 = (3^x + 3^{-x})(9^x - 3^x \cdot 3^{-x} + 9^{-x}) = 4(14 - 1) = 52

.

The identity

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

allows factoring cubic exponential expressions.

3

Substitute the evaluated terms into the target rational expression and simplify.

\frac{27^x + 27^{-x} - 2}{9^x + 9^{-x} + 1} = \frac{52 - 2}{14 + 1} = \frac{50}{15} = \frac{10}{3}.

Replacing component expressions with their computed values yields the simplified numerical fraction.

Key Concept

Evaluating high-power exponential expressions using polynomial identity transformations and exponent laws.

Question 1933Question →

Far from being an intentional __________ of the empirical record, the climatologist’s understated projections regarding sea-level rise reflected a conservative methodology designed to minimize speculative risk rather than a deliberate attempt to mislead the public. Which two of the following answer choices, when inserted into the sentence, produce completed sentences that are identical in meaning?

Select all that apply

distortion

exaggeration

falsification

hyperbole

revision

precision

Show answer & explanation

Answer: distortion; falsification

Answer

distortion and falsification

The sentence specifies that the climatologist's understated projections were not a 'deliberate attempt to mislead the public.' Therefore, the blank requires words meaning intentional misrepresentation or alteration of facts. The words 'distortion' and 'falsification' both accurately convey this meaning, and substituting either into the sentence yields completed sentences with identical meaning.

Step-by-Step Solution

1

Analyze structural clues and contrast signals in the sentence stem.

The phrase 'rather than a deliberate attempt to mislead' indicates that the blank must mean a deliberate misrepresentation, altering, or faking of data.

The contrast structure 'Far from being an intentional [blank]... reflected a conservative methodology... rather than a deliberate attempt to mislead' demands a word denoting deceptive alteration of facts.

2

Evaluate option pairs for semantic equivalence in context.

'distortion' and 'falsification' both mean altering or misrepresenting empirical data to mislead, creating equivalent sentence meanings.

Both words fit the contextual requirement of dishonest data manipulation and produce sentences with identical overall meaning.

3

Identify false synonym traps and topically related distractors.

'exaggeration' and 'hyperbole' form a valid synonym pair but fail contextually because the projections were 'understated' (overstatement contradicts the context). 'revision' and 'precision' are topically relevant scientific terms but lack synonymous partners.

Eliminating false synonym pairs that fail context and unpaired topical distractors yields the unique correct answer pair.

Key Concept

Eliminating False Synonyms and Topically Related Distractors

Question 1934Question →

Two positive integers $x$ and $y$ satisfy $x < y$ , $\gcd(x, y) = 18$ , and $\text{lcm}(x, y) = 1080$ . If $x$ has exactly $6$ positive divisors, what is the value of $y$ ?

216

270

360

540

1080

Show answer & explanation

Answer: $1080$

Answer

1080

Since

\gcd(x, y) = 18

and

\text{lcm}(x, y) = 1080

, we can express

x = 18m

and

y = 18n

with

\gcd(m, n) = 1

and

m < n

. Substituting into

\text{lcm}(x, y) = 18mn = 1080

gives

mn = 60

. The coprime pairs

(m, n)

with

m < n

are

(1, 60)

,

(3, 20)

,

(4, 15)

, and

(5, 12)

. Testing

x = 18m

for each pair: for

m = 1

,

x = 18 = 2^1 \cdot 3^2

, which has

(1+1)(2+1) = 6

positive divisors. This matches the condition, so

n = 60

and

y = 18 \times 60 = 1080

.

Step-by-Step Solution

1

Express

x

and

y

in terms of their Greatest Common Divisor

Let

x = 18m

and

y = 18n

, where

m

and

n

are positive integers such that

m < n

and

\gcd(m, n) = 1

.

Since

\gcd(x, y) = 18

, both numbers must be multiples of

18

, and their remaining factor parts must be coprime.

2

Use the LCM formula

\text{lcm}(x, y) = 18mn

to find

mn

18mn = 1080 \implies mn = 60

.

The least common multiple of

18m

and

18n

with

\gcd(m,n)=1

is

18mn

.

3

Find all coprime factor pairs

(m, n)

of

60

with

m < n

The prime factorization of

60

is

2^2 \cdot 3^1 \cdot 5^1

. Coprime pairs

(m, n)

are

(1, 60)

,

(3, 20)

,

(4, 15)

, and

(5, 12)

.

Pairs must be coprime so that

\gcd(18m, 18n) = 18

remains true.

4

Determine the number of positive divisors for

x = 18m

for each candidate pair

For

m = 1

,

x = 18 = 2^1 \cdot 3^2

, which has

(1+1)(2+1) = 6

positive divisors.

The number of positive divisors of

p_1^{a_1} p_2^{a_2} \dots

is given by

(a_1 + 1)(a_2 + 1) \dots

. Only

m = 1

gives exactly

6

divisors.

5

Calculate

y

y = 18n = 18 \cdot 60 = 1080

.

Corresponding to

m = 1

,

n = 60

gives

y = 1080

.

Key Concept

Prime Factorization, GCD/LCM Coprime Relationships, and Divisor Counting Formula

Alternative Method

Alternatively, use

\gcd(x, y) \cdot \text{lcm}(x, y) = x \cdot y \implies x \cdot y = 18 \cdot 1080 = 19440

. Since

x

has

6

positive divisors and

x

is a multiple of

18

,

x

must equal

18

because

18 = 2^1 \cdot 3^2

has

(1+1)(2+1) = 6

divisors. Then

y = 19440 / 18 = 1080

.

Estimated Time:1m 30s

Question 1935Question →

In trapezoid $ABCD$ , side $AB$ is parallel to side $CD$ , and $\angle DAB = 90^\circ$ . If $AB = 14$ , $CD = 8$ , and $BC = 10$ , what is the area of trapezoid $ABCD$ ?

40

66

88

110

176

Show answer & explanation

Answer: 88

Answer

88

By drawing an altitude from point

C

perpendicular to base

AB

at point

H

, the trapezoid is partitioned into a rectangle

ADCH

and a right triangle

CHB

. The base of triangle

CHB

is

HB = 14 - 8 = 6

, and its hypotenuse is

BC = 10

. Applying the Pythagorean theorem yields

CH = \sqrt{10^2 - 6^2} = 8

. Using the trapezoid area formula

\frac{b_1 + b_2}{2} \times h

gives

\frac{14 + 8}{2} \times 8 = 88

.

Step-by-Step Solution

1

Decompose the trapezoid into a rectangle and a right triangle.

Draw a line segment from vertex

C

perpendicular to base

AB

, intersecting

AB

at point

H

. Quadrilateral

ADCH

is a rectangle, and triangle

CHB

is a right triangle with right angle at

H

.

Splitting the trapezoid allows the perpendicular height to be determined using right triangle properties.

2

Determine the horizontal base of right triangle

CHB

.

Since

ADCH

is a rectangle,

AH = CD = 8

. Therefore,

HB = AB - AH = 14 - 8 = 6

.

Opposite sides of a rectangle are equal in length.

3

Calculate height

CH

using the Pythagorean theorem.

In right triangle

CHB

,

CH^2 + HB^2 = BC^2 \implies CH^2 + 6^2 = 10^2 \implies CH^2 + 36 = 100 \implies CH = \sqrt{64} = 8

.

The height of the trapezoid corresponds to leg

CH

of right triangle

CHB

.

4

Calculate the area of trapezoid

ABCD

.

\text{Area} = \frac{AB + CD}{2} \times CH = \frac{14 + 8}{2} \times 8 = 11 \times 8 = 88.

The area formula for a trapezoid is half the sum of the parallel bases multiplied by the perpendicular height.

Key Concept

Decomposing a trapezoid into a rectangle and a right triangle to calculate height via the Pythagorean theorem

Estimated Time:1m 15s

Question 1936Question →

A technology startup recorded the monthly software subscription costs, in dollars, for 6 department teams: $450, 120, 310, 120, 280,$ and $500$ . A seventh department team with a monthly subscription cost of $x$ dollars is added to the dataset, where $x$ is a positive integer. Which of the following statements regarding the measures of central tendency for the updated 7-team dataset must be true? Select all that apply.

Select all that apply

The median of the subscription costs for the 7 teams cannot be greater than 310.

If

x = 600

, the arithmetic mean of the subscription costs for the 7 teams is greater than the median.

If

x = 120

, the median of the subscription costs for the 7 teams is 120.

The arithmetic mean of the subscription costs for the 7 teams will equal the median if

x = 290

.

The mode of the subscription costs for the 7 teams is uniquely 120 for any positive integer value of

x

.

Show answer & explanation

Answer: The median of the subscription costs for the 7 teams cannot be greater than 310.; If $x = 600$ , the arithmetic mean of the subscription costs for the 7 teams is greater than the median.

Answer

The statements that must be true are that the median cannot exceed 310, and that if x = 600, the arithmetic mean is greater than the median.

Sorting the six known values gives 120, 120, 280, 310, 450, 500. In a dataset of 7 numbers, the median is the 4th value when sorted. If x is added, the 4th value will be 280 if x ≤ 280, x if 280 < x < 310, or 310 if x ≥ 310. Thus, the median is capped at 310. Furthermore, setting x = 600 yields a median of 310 and a mean of (1780 + 600) / 7 = 340, which is greater than the median.

Step-by-Step Solution

1

Sort the existing 6 dataset elements in ascending order and compute their sum.

Sorted list: 120, 120, 280, 310, 450, 500. Sum = 1780.

Establishing the baseline position of values helps determine how inserting x shifts the median, mean, and mode.

2

Analyze the position of the median for a 7-element dataset.

The median is the 4th element when sorted. If x ≤ 280, the 4th element is 280. If 280 < x < 310, the 4th element is x. If x ≥ 310, the 4th element is 310. Thus, 280 ≤ median ≤ 310.

This establishes upper and lower bounds for the median regardless of x.

3

Evaluate the statement regarding median upper bound (median ≤ 310).

Since the maximum possible 4th element in the sorted 7-element list is 310, the median can never be greater than 310.

Confirms the validity of the statement concerning the maximum median.

4

Evaluate the scenario where x = 600.

Sum = 1780 + 600 = 2380. Mean = 2380 / 7 = 340. Median = 310 (since 600 ≥ 310). 340 > 310.

Confirms that when x = 600, mean (340) is strictly greater than median (310).

Key Concept

Measures of Central Tendency (Mean, Median, Mode)

Question 1937Question →

In Year 1, a technology firm allocated $40\%$ of its total operating budget to Research and Development (R&D) and the remaining $60\%$ to Business Operations. In Year 2, the amount allocated to R&D was increased by $25\%$ over its Year 1 amount, while the amount allocated to Business Operations was decreased by $15\%$ from its Year 1 amount. If the firm's total operating budget in Year 2 was $\$505,000$ , what was its total operating budget in Year 1?

\$454,500

\$480,952

\$500,000

\$499,950

\$510,050

Show answer & explanation

Answer: $\$500,000$

Answer

The total operating budget in Year 1 was

\$500,000

.

The budget allocation in Year 1 assigns

40\%

to R&D and

60\%

to Operations. In Year 2, R&D becomes

0.40 \times 1.25 = 0.50

of the Year 1 budget, and Operations becomes

0.60 \times 0.85 = 0.51

of the Year 1 budget. Adding these yields a Year 2 total equal to

1.01

times the Year 1 budget. Setting

1.01 \times \text{Year 1 Budget} = \$505,000

and dividing gives

\$500,000

.

Step-by-Step Solution

1

Define Year 1 components in terms of total Year 1 budget

B

.

R&D allocation =

0.40B

; Operations allocation =

0.60B

.

The problem specifies a

40\% / 60\%

split of the total budget

B

in Year 1.

2

Calculate Year 2 allocations after percentage changes.

Year 2 R&D =

0.40B \times 1.25 = 0.50B

; Year 2 Operations =

0.60B \times 0.85 = 0.51B

.

R&D increases by

25\%

(

1 + 0.25 = 1.25

) and Operations decreases by

15\%

(

1 - 0.15 = 0.85

).

3

Sum Year 2 allocations to express total Year 2 budget in terms of

B

.

Total Year 2 Budget =

0.50B + 0.51B = 1.01B

.

Combining both component allocations gives the total budget for Year 2.

4

Solve for the Year 1 budget

B

using the given Year 2 total.

1.01B = \$505,000 \implies B = \frac{505,000}{1.01} = \$500,000

.

Dividing the Year 2 budget by the total growth multiplier yields the original Year 1 budget.

Key Concept

Weighted Percent Change and Initial Base Calculation

Question 1938Question →

In the $xy$ -plane, line $k$ passes through the points $(-6, 0)$ and $(0, 4)$ . Line $m$ is perpendicular to line $k$ and intersects line $k$ at its $y$ -intercept. What is the $x$ -intercept of line $m$ ?

-\frac{8}{3}

-6

\frac{3}{2}

\frac{8}{3}

6

Show answer & explanation

Answer: $\frac{8}{3}$

Answer

The

x

-intercept of line

m

is

\frac{8}{3}

.

Line

k

has a slope of

\frac{4 - 0}{0 - (-6)} = \frac{2}{3}

. Because line

m

is perpendicular to line

k

, its slope is the negative reciprocal,

-\frac{3}{2}

. Line

m

intersects line

k

at its

y

-intercept

(0, 4)

, giving line

m

the equation

y = -\frac{3}{2}x + 4

. Setting

y = 0

yields

0 = -\frac{3}{2}x + 4

, which solves to

x = \frac{8}{3}

.

Step-by-Step Solution

1

Calculate the slope of line

k

.

The slope

m_k = \frac{4 - 0}{0 - (-6)} = \frac{4}{6} = \frac{2}{3}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is calculated using

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

.

2

Determine the slope and

y

-intercept of line

m

.

The slope of line

m

is

m_m = -\frac{3}{2}

, and its

y

-intercept is

(0, 4)

.

Perpendicular lines have slopes that are negative reciprocals of each other, and line

m

intersects line

k

at

(0, 4)

.

3

Write the equation of line

m

and solve for its

x

-intercept.

Setting

y = 0

in

y = -\frac{3}{2}x + 4

yields

0 = -\frac{3}{2}x + 4 \implies \frac{3}{2}x = 4 \implies x = \frac{8}{3}

.

The

x

-intercept of a line is the value of

x

when

y = 0

.

Key Concept

Perpendicular line slopes and intercept calculations

Question 1939Question →

Passage:
In 1856, American scientist Eunice Newton Foote presented a paper demonstrating the heat-trapping capabilities of atmospheric gases. Using glass cylinders exposed to sunlight, Foote measured the temperature changes in containers filled with compressed air, hydrogen, oxygen, and carbon dioxide. Her measurements revealed that the cylinder containing moist carbon dioxide experienced the highest temperature increase and took the longest duration to cool down after being removed from sunlight. Based on these observations, Foote explicitly posited that an atmosphere enriched with carbon dioxide would impart a significantly higher temperature to Earth. Although John Tyndall is frequently credited with discovering the atmospheric greenhouse effect through his later 1859 experiments using infrared spectrophotometry, Foote’s earlier empirical work established the direct relationship between carbon dioxide concentrations and solar radiative heating, despite her apparatus omitting isolated infrared radiation sources.

Which of the following statements about Eunice Newton Foote's 1856 experiment are explicitly supported by the passage? Select all that apply.

Select all that apply

Foote recorded temperature changes in glass cylinders containing several different gases, including hydrogen and carbon dioxide.

Foote used infrared spectrophotometry to measure how different gas samples reacted to solar radiation.

The container filled with moist carbon dioxide required a longer period to cool down than containers holding the other tested gases.

Show answer & explanation

Answer: Foote recorded temperature changes in glass cylinders containing several different gases, including hydrogen and carbon dioxide.; The container filled with moist carbon dioxide required a longer period to cool down than containers holding the other tested gases.

Answer

The statements confirming that Foote recorded temperatures across multiple gases (including hydrogen and carbon dioxide) and that the moist carbon dioxide container required the longest time to cool down are both explicitly supported by the passage.

The passage directly supports two statements: it lists hydrogen and carbon dioxide among the gases tested in Foote's glass cylinders, and it explicitly states that the container holding moist carbon dioxide took the longest duration to cool down compared to the other tested containers.

Step-by-Step Solution

1

Locate the sentence detailing the specific gases placed in the glass cylinders during Foote's experiment.

The text explicitly names compressed air, hydrogen, oxygen, and carbon dioxide.

Verifies whether testing hydrogen and carbon dioxide is directly stated.

2

Examine the passage for statements regarding the cooling duration of the moist carbon dioxide container.

The text explicitly states that the cylinder containing moist carbon dioxide 'took the longest duration to cool down after being removed from sunlight.'

Verifies whether the relative cooling time claim is explicitly supported.

3

Evaluate the statement attributing infrared spectrophotometry to Foote's methodology.

The passage states that infrared spectrophotometry was introduced by John Tyndall in 1859 and explicitly mentions that Foote's apparatus omitted isolated infrared radiation sources.

Eliminates statements that misread or misattribute passage details.

Key Concept

Explicit Detail Retrieval

Question 1940Question →

A venture capital fund allocated an initial sum of money between Portfolio Alpha and Portfolio Beta, with Portfolio Alpha receiving $60\%$ of the total sum and Portfolio Beta receiving the remaining $40\%$ . During the first year, the value of Portfolio Alpha increased by $20\%$ , while the value of Portfolio Beta decreased by $15\%$ . During the second year, the value of Portfolio Alpha decreased by $10\%$ relative to its value at the end of the first year, while the value of Portfolio Beta increased by $25\%$ relative to its value at the end of the first year. By what percent did the total combined value of the two portfolios increase from the initial allocation to the end of the second year?

Show answer & explanation

Answer: 7.3

Answer

The total combined value of the two portfolios increased by 7.3%.

To find the net percent change over the two-year period, track each portfolio's value year by year. Assuming an initial combined total of

100, Portfolio Alpha begins at

60 and Portfolio Beta at

40. By the end of Year 1, Alpha increases by 20% to

72 (

60 \times 1.20

), while Beta decreases by 15% to

34 (

40 \times 0.85

). In Year 2, Alpha decreases by 10% relative to its Year 1 ending value, becoming

64.80 (

72 \times 0.90

), while Beta increases by 25% relative to its Year 1 ending value, becoming

42.50 (

34 \times 1.25

). The total combined value at the end of Year 2 is

64.80 +

42.50 =

107.30. Relative to the initial $100 allocation, this represents a net increase of 7.3%.

Step-by-Step Solution

1

Assign a convenient base value for the initial total investment.

Assume an initial total allocation of

100. Portfolio Alpha starts with

60 and Portfolio Beta starts with $40.

Percent changes are proportional and scale-invariant, making $100 a simple base value for calculation.

2

Compute the value of each portfolio at the end of Year 1.

Portfolio Alpha =

60 \times (1 + 0.20) =

72. Portfolio Beta =

40 \times (1 - 0.15) =

34.

Apply Year 1 growth (+20%) and loss (-15%) to their respective starting funds.

3

Compute the value of each portfolio at the end of Year 2.

Portfolio Alpha =

72 \times (1 - 0.10) =

64.80. Portfolio Beta =

34 \times (1 + 0.25) =

42.50.

Apply Year 2 changes (-10% and +25%) to the Year 1 ending values rather than the original principal.

4

Find the final total combined value and calculate the overall net percentage increase.

Combined Year 2 value =

64.80 +

42.50 = $107.30. Net percentage increase = \frac{107.30 - 100}{100} \times 100\% = 7.3\%.

Compare the final combined sum of

107.30 to the initial combined sum of

100.

Key Concept

Successive percent changes with shifting base values across multiple assets

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

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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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

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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