All practice questions

Question 941Question →

The passage below is adapted from a book on the history of mathematics. Which punctuation mark should be placed in the blank to correctly introduce the explanation of the two faculties?

Fill in the blanks below

In his 1912 treatise on the philosophy of science, Henri Poincaré argued that mathematical discoveries are driven by two distinct cognitive faculties logical analysis, which provides rigor and certainty, and creative intuition, which enables synthesis and invention.

Show answer & explanation

Answer

A colon or a dash (such as an em dash)

A colon or a dash is correct because the clause preceding the blank is independent ('In his 1912 treatise... cognitive faculties') and the text following the blank is an explanation or list that elaborates on the 'two distinct cognitive faculties' mentioned. A colon or a single dash is standardly used to separate an independent clause from an explanatory list or phrase.

Step-by-Step Solution

1

Analyze the grammatical structure of the clause preceding the blank.

The clause 'In his 1912 treatise on the philosophy of science, Henri Poincaré argued that mathematical discoveries are driven by two distinct cognitive faculties' is independent because it contains a subject ('Henri Poincaré') and a verb ('argued') and can stand alone as a complete sentence.

Identifying whether the preceding clause is independent is necessary to determine which punctuation marks (such as colons, semicolons, or dashes) are grammatically permissible.

2

Analyze the grammatical structure of the phrase following the blank.

The phrase 'logical analysis, which provides rigor and certainty, and creative intuition, which enables synthesis and invention' is a list of two non-independent noun phrases with nonessential clauses attached to them.

Identifying the structure of the following phrase determines the relationship between the two parts of the sentence and rules out punctuation marks that require an independent clause on both sides.

3

Choose the appropriate punctuation mark to link the independent clause to the explanatory list.

A colon or a dash is correct because both can join an independent clause to a list or explanation. A semicolon is incorrect because a semicolon must separate two independent clauses.

Applying standard English punctuation rules ensures that the transition between the independent clause and the elaboration is grammatically sound.

Key Concept

Colons and Dashes

Question 942Question →

A water tank contains 24 gallons of water. Water is being drained from the tank at a constant rate. After 8 minutes, the tank contains 14 gallons of water. If the volume of water in the tank, in gallons, is a linear function of the time in minutes, how many minutes will it take for the tank to contain exactly 4 gallons of water?

Show answer & explanation

Answer: 16

Answer

16

The volume of water in the tank decreases linearly from an initial value of

24

gallons at

t = 0

to

14

gallons at

t = 8

. The constant rate of change (slope) is calculated by dividing the change in volume by the change in time:

\frac{14 - 24}{8 - 0} = -1.25

gallons per minute. Using the slope-intercept form, the volume

V

at time

t

is given by

V = -1.25t + 24

. Setting the volume

V = 4

gives the equation

4 = -1.25t + 24

. Solving for

t

yields

-20 = -1.25t

, which simplifies to

t = 16

.

Step-by-Step Solution

1

Determine the initial state and the state after 8 minutes as coordinate points.

The initial state is

(0, 24)

and the state after

8

minutes is

(8, 14)

.

These coordinates represent the relationship between time and volume of water in the tank.

2

Calculate the slope (constant rate of change) of the linear function.

Slope

m = \frac{14 - 24}{8 - 0} = -1.25

The slope represents the constant rate at which water is being drained from the tank.

3

Formulate the linear equation.

V(t) = -1.25t + 24

Using the slope-intercept form

V(t) = mt + b

, where

b = 24

is the vertical intercept representing the initial volume.

4

Solve for the time

t

when the volume of water is

4

gallons.

4 = -1.25t + 24 \implies -20 = -1.25t \implies t = 16

To find the time at which the volume decreases to exactly

4

gallons.

Key Concept

Linear Functions and Graphs

Question 943Question →

Launched in 1977 to study the outer Solar System, the Voyager 1 spacecraft has traveled farther from Earth than any other human-made object. By the time the probe, which carries a golden record containing sounds of Earth, finally reached interstellar space in 2012, it ______ telemetry data back to NASA scientists for thirty-five years. Which choice completes the text so that it conforms to the conventions of Standard English?

has been transmitting

had been transmitting

was transmitting

will have transmitted

Show answer & explanation

Answer: had been transmitting

Answer

had been transmitting

The passage establishes a specific point in the past ('reached interstellar space in 2012') and describes an action that started prior to this point and was ongoing up to it ('for thirty-five years'). To represent this continuous action leading up to a past reference point, the past perfect progressive tense ('had been transmitting') is required.

Step-by-Step Solution

1

Identify the temporal reference point established in the sentence.

The reference point is 'reached interstellar space in 2012,' which is in the past.

Establishing the timeline helps determine the correct verb tense.

2

Determine the relationship between the blank's action and the reference point.

The action of sending data ('telemetry data... for thirty-five years') was ongoing and completed up to that past point in time.

An action that was ongoing prior to and up to a past reference point requires the past perfect progressive tense.

Key Concept

Past Perfect Progressive Tense

Estimated Time:1m 0s

Question 944Question →

Complete the text by filling in the blank with the appropriate subject pronoun or noun phrase that conforms to the conventions of Standard English subject-modifier agreement.

Fill in the blanks below

In 1928, Paul Dirac formulated a relativistic wave equation to describe the behavior of the electron. Though initially dismissed by contemporary physicists as a mathematical curiosity with no physical relevance, is now recognized as a foundational equation in quantum mechanics.

Show answer & explanation

Answer

The blank must be filled with a noun phrase or pronoun that refers directly to the equation (such as 'the Dirac equation', 'this equation', or 'it') to ensure it agrees with the introductory modifying phrase 'Though initially dismissed by contemporary physicists as a mathematical curiosity with no physical relevance.'

The introductory modifying phrase 'Though initially dismissed by contemporary physicists as a mathematical curiosity with no physical relevance' must logically describe the subject that immediately follows it. In this context, the entity that was dismissed as a mathematical curiosity is the relativistic wave equation. Therefore, the blank must be filled with a noun phrase or pronoun referring to the equation (such as 'the Dirac equation' or 'it') to establish correct subject-modifier agreement.

Step-by-Step Solution

1

Identify the introductory modifying phrase and determine what it logically describes.

The modifying phrase is 'Though initially dismissed by contemporary physicists as a mathematical curiosity with no physical relevance.' The entity that was dismissed as a 'mathematical curiosity' is the relativistic wave equation formulated by Paul Dirac, not Paul Dirac himself or the contemporary physicists.

According to the rules of subject-modifier agreement, the subject of the main clause must immediately follow the introductory modifier and must be the entity that is logically described by that modifier.

2

Select a subject noun phrase or pronoun that refers to the equation to fill the blank.

Noun phrases like 'the Dirac equation', 'this equation', or the pronoun 'it' correctly identify the equation as the subject of the main clause.

This alignment prevents a dangling modifier and ensures that the sentence is grammatically correct and coherent.

Key Concept

Subject-Modifier Agreement

Question 945Question →

If $5(x - 3) - 2x = 9$ , what is the value of $x$ ?

2

4

8

-2

Show answer & explanation

Answer: 8

Answer

The value of

x

is

8

.

To solve the equation, distribute the 5 to the terms inside the parentheses to get

5x - 15 - 2x = 9

. Combine the like terms

5x

and

-2x

to simplify the equation to

3x - 15 = 9

. Add 15 to both sides to isolate the variable term, which yields

3x = 24

. Finally, divide both sides by 3 to find

x = 8

.

Step-by-Step Solution

1

Distribute the

5

to the terms inside the parentheses.

5x - 15 - 2x = 9

To remove the parentheses and begin simplifying the equation.

2

Combine the like variable terms

5x

and

-2x

.

3x - 15 = 9

To group the x terms together.

3

Add

15

to both sides of the equation.

3x = 24

To isolate the term with the variable on one side.

4

Divide both sides of the equation by

3

.

x = 8

To solve for

x

.

Key Concept

Solving linear equations in one variable using the distributive property, combining like terms, and applying inverse operations.

Estimated Time:45s

Question 946Question →

Complete the academic passage below by writing the correct form of the verb 'champion' in the blank. What is the correct form of the verb 'champion' that completes the sentence in accordance with the conventions of Standard English?

Fill in the blanks below

By the time French physicist Émilie du Châtelet completed her definitive translation of Isaac Newton's *Principia* in 1749, continental mathematicians René Descartes's vortex theory of planetary motion for over a century. Consequently, Châtelet’s extensive mathematical commentary, which demonstrated the compatibility of Newtonian mechanics with Leibnizian calculus, played a crucial role in shifting the scientific consensus of Europe.

Show answer & explanation

Answer

had championed (or had been championing)

The correct verb form is the past perfect tense ('had championed') or past perfect continuous ('had been championing'). The opening dependent clause uses 'By the time' to anchor the sentence's perspective to a point in the past (1749). Because the action in the main clause occurs over a duration ('for over a century') prior to this point, the past perfect is required to establish the correct temporal hierarchy.

Step-by-Step Solution

1

Analyze the temporal markers in the sentence to determine the timeline of events.

The sentence establishes a reference point in the past ('completed... in 1749') using the phrase 'By the time.' The action in the main clause occurs over a duration ('for over a century') leading up to that point.

Determining the relationship between the past reference point and the preceding continuous action dictates the correct grammatical aspect.

2

Formulate the correct tense and aspect of the verb.

An action that occurred over a duration prior to a specific point in the past requires the past perfect ('had championed') or the past perfect continuous ('had been championing').

This shows the correct sequence of events, indicating that the championing of Descartes's theory happened before Châtelet completed her translation.

Key Concept

Verb Tense, Aspect, and Mood

Estimated Time:1m 30s

Question 947Question →

The passage below describes the scientific investigation of the 1908 Tunguska event. Which form of the verb 'elapse' correctly completes the sentence to conform to the conventions of Standard English?

Fill in the blanks below

In 1908, a massive explosion occurred near the Podkamennaya Tunguska River in Russia, flattening millions of trees. Because the region was extremely remote and politically unstable, scientific research was delayed for years. By the time the first scientific expedition finally reached the site in 1927, nearly two decades since the event, leaving researchers to study decaying tree trunks rather than fresh evidence.

Show answer & explanation

Answer

had elapsed

The sentence establishes a timeline where the elapsing of two decades occurred before another past event (the first scientific expedition reaching the site in 1927). Therefore, the past perfect tense 'had elapsed' is required to show that one past action was completed prior to another past action.

Step-by-Step Solution

1

Identify the temporal framework established by the surrounding context.

The passage describes past events, establishing a timeline in the past: the explosion occurred in 1908, and the first expedition reached the site in 1927 (indicated by the simple past verb 'reached').

Understanding the timeline helps determine whether to use past, present, or future tense.

2

Analyze the relationship between the two past events to determine the correct aspect.

The phrase 'By the time...' indicates that the elapsing of the two decades was completed before the expedition arrived in 1927.

An action that is completed prior to a specific point in the past requires the past perfect tense ('had' + past participle).

3

Apply the correct conjugation of the verb 'elapse'.

The past perfect form of 'elapse' is 'had elapsed'.

This correctly conforms to the conventions of Standard English for representing a past action that precedes another past action.

Key Concept

Using the past perfect tense ('had' + past participle) to describe an action that was completed before another action in the past.

Estimated Time:1m 0s

Question 948Question →

The mimic octopus ______ possesses an extraordinary ability to impersonate other marine species. By dynamically altering its shape, color, and movement, this clever cephalopod can mimic creatures such as flatfish, lionfish, and sea snakes. Marine biologists have observed that the animal adjusts its behavior depending on the specific threat it faces, demonstrating an impressive level of situational awareness.

Which choice completes the text so that it conforms to the conventions of Standard English?

, a unique cephalopod native to the Indo-Pacific,

, a unique cephalopod native to the Indo-Pacific—

—a unique cephalopod native to the Indo-Pacific,

a unique cephalopod native to the Indo-Pacific,

Show answer & explanation

Answer: , a unique cephalopod native to the Indo-Pacific,

Answer

The option that correctly uses a matching pair of commas: ', a unique cephalopod native to the Indo-Pacific,'

The phrase 'a unique cephalopod native to the Indo-Pacific' is a nonessential appositive that provides extra information about the subject 'The mimic octopus.' Because the main clause remains grammatically complete without it, this nonessential element must be set off from the rest of the sentence using a matching pair of punctuation marks. The correct option uses a matching pair of commas (one before the phrase and one after it) to correctly set off the nonessential element.

Step-by-Step Solution

1

Identify the subject of the sentence and the core clause to determine if the intervening phrase is essential or nonessential.

The core clause is 'The mimic octopus possesses an extraordinary ability...' which is grammatically complete on its own. The phrase 'a unique cephalopod native to the Indo-Pacific' is a nonessential appositive.

Nonessential elements must be set off from the rest of the sentence with matching punctuation marks.

2

Analyze the punctuation surrounding the nonessential phrase in each option to look for matching pairs.

Only the option using two commas provides a matched pair on both sides of the phrase. Other options combine different marks (a comma and an em dash, or an em dash and a comma) or omit one of the marks entirely.

Standard English conventions require that nonessential elements be set off with symmetrical punctuation (comma-comma, dash-dash, or parenthesis-parenthesis).

Key Concept

Setting off nonessential elements with symmetrical punctuation marks (matching commas, em dashes, or parentheses).

Estimated Time:45s

Question 949Question →

The passage below contains a blank indicating a missing punctuation mark. Which punctuation mark should be placed in the blank to correctly set off the nonessential relative clause?

Fill in the blanks below

Discovered in 1799 near the Egyptian town of Rashid, the Rosetta Stone has played a vital role in modern linguistics. The ancient slab, which features three translations of a single decree allowed scholars to decipher Egyptian hieroglyphs. By comparing the Greek text with the hieroglyphic version, researchers finally unlocked the mysteries of an ancient writing system.

Show answer & explanation

Answer

A comma (`,`) must be placed in the blank.

The clause 'which features three translations of a single decree' is a nonessential relative clause that provides descriptive detail about 'the ancient slab.' To set off nonessential elements, matching punctuation marks must be used at the beginning and end of the element. Since a comma is used after 'slab' to open the clause, a matching comma must be placed in the blank to close the clause.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The clause 'which features three translations of a single decree' is a nonessential relative clause because it provides extra information about 'the ancient slab' but is not required to identify the slab or complete the main idea of the sentence.

Recognizing nonessential clauses helps determine if and how they should be set off with punctuation.

2

Check the punctuation already used to introduce the nonessential clause.

The clause begins with a comma after 'slab'.

This establishes the punctuation style used for the sentence.

3

Apply the rule of punctuation symmetry to complete the sentence.

A matching comma must be placed in the blank to close the nonessential clause.

Nonessential elements must be set off by matching punctuation marks (comma-comma, dash-dash, or parenthesis-parenthesis).

Key Concept

Punctuation of Nonessential Elements

Question 950Question →

Theoretical models of planetary formation often emphasize the initial abundance of heavy elements in a protoplanetary disk. However, in analyzing the accretion rates of Jovian planets, astrophysicist Elena Vance argues that a nascent world's ultimate size depends not on how rapidly it accumulates local gas and dust from the disk during its early formation phase, but on ______

Which choice completes the passage so that it conforms to the conventions of Standard English?

how efficiently it clears nearby planetesimals from its orbital pathway.

clearing nearby planetesimals from its orbital pathway efficiently.

the efficiency with which nearby planetesimals are cleared from its orbital pathway.

whether it can efficiently clear nearby planetesimals from its orbital pathway.

Show answer & explanation

Answer: how efficiently it clears nearby planetesimals from its orbital pathway.

Answer

The correct answer is the clause that mirrors the structure of the first element by starting with the conjunction 'how' followed by an adverb, subject pronoun, and active verb: 'how efficiently it clears nearby planetesimals from its orbital pathway.'

The correct option is correct because it uses a dependent clause that perfectly mirrors the grammatical structure of the first clause in the comparison. The first clause, 'how rapidly it accumulates...', is structured as a subordinating conjunction ('how') followed by an adverb ('rapidly'), a subject ('it'), and an active verb ('accumulates'). The correct option follows this exact pattern: 'how' + adverb ('efficiently') + subject ('it') + active verb ('clears'). This maintains a balanced and parallel sentence structure.

Step-by-Step Solution

1

Identify the comparative structure in the sentence.

The sentence uses the correlative comparison 'depends not on [X], but on [Y]'.

Parallel structure requires that X and Y have matching grammatical forms after the preposition 'on'.

2

Analyze the grammatical structure of the first element (X).

The first element is 'how rapidly it accumulates local gas and dust...', which is a noun clause structured as: conjunction ('how') + adverb ('rapidly') + subject ('it') + active verb ('accumulates') + object ('local gas and dust') + modifiers.

To maintain parallelism, the second element (Y) must follow this exact same grammatical pattern.

3

Evaluate the choices to find the one that matches this pattern.

The option containing 'how efficiently it clears nearby planetesimals from its orbital pathway' uses the exact same pattern: conjunction ('how') + adverb ('efficiently') + subject ('it') + active verb ('clears') + object ('nearby planetesimals') + modifiers.

Matching grammatical structure ensures the sentence is parallel and grammatically correct.

Key Concept

Parallel Structure

Question 951Question →

In his eleventh-century treatise *Book of Optics*, polymath Ibn al-Haytham demonstrated that light travels in straight lines through his experiments with a dark chamber, or camera obscura. Although earlier scholars ______ similar optical phenomena, al-Haytham was the first to provide a systematic, mathematical explanation of how light rays project inverted images onto a screen.

Which choice completes the text so that it conforms to the conventions of Standard English?

observe

have observed

had observed

will observe

Show answer & explanation

Answer: had observed

Answer

had observed

The past perfect verb 'had observed' is correct because the passage describes events that occurred in the past (using the past tense verbs 'demonstrated' and 'was'). Since the actions of the 'earlier scholars' occurred before Ibn al-Haytham's eleventh-century discoveries, the past perfect tense is required to establish the correct temporal order of these past events.

Step-by-Step Solution

1

Identify the temporal framework of the passage by examining the surrounding verbs.

The verbs 'demonstrated' and 'was' are in the simple past tense, establishing that the narrative is set in the past.

Understanding the primary time frame is necessary to select a verb form that is logically consistent.

2

Determine the chronological relationship between the action in the blank and the primary time frame.

The blank describes the actions of 'earlier scholars' which occurred prior to Ibn al-Haytham's eleventh-century work.

Identifying that an action occurred before another past event requires the use of the past perfect tense ('had' + past participle).

3

Evaluate the options to find the form that represents the past perfect tense.

The verb phrase 'had observed' is in the past perfect tense, which fits the context perfectly.

Selecting the grammatically correct option ensures tense consistency and logical flow.

Key Concept

Verb Tense, Aspect, and Mood

Question 952Question →

Deep-sea hydrothermal vents support thriving ecosystems despite extreme temperatures and pressures. The vent shrimp *Rimicaris exoculata* _______ relies on a specialized dorsal organ to detect the faint, near-infrared glow emitted by the vents. This unique adaptation allows the species to navigate its hostile environment without the benefit of traditional eyesight.

Which choice completes the text so that it conforms to the conventions of Standard English?

—which lacks functional compound eyes,

, which lacks functional compound eyes,

, which lacks functional compound eyes—

which lacks functional compound eyes,

Show answer & explanation

Answer: , which lacks functional compound eyes,

Answer

The choice that sets off the nonessential relative clause with matching commas: ', which lacks functional compound eyes,'

The sentence contains the nonessential relative clause 'which lacks functional compound eyes' modifying the specific species name *Rimicaris exoculata*. Because the species is already uniquely identified by its scientific name, this description is nonessential and must be set off from the main clause with matching punctuation. The correct option uses a pair of commas to correctly isolate this clause.

Step-by-Step Solution

1

Analyze the sentence structure to identify the relationship between the clause 'which lacks functional compound eyes' and the noun phrase 'Rimicaris exoculata'.

The clause is nonessential (non-restrictive) because 'Rimicaris exoculata' is a specific scientific name that uniquely identifies the organism.

Nonessential elements provide extra information and must be set off from the rest of the sentence with punctuation.

2

Determine the required punctuation rules for nonessential elements.

The punctuation setting off the nonessential element must be symmetrical (matching pairs: comma-comma, dash-dash, or parenthesis-parenthesis).

Mismatched punctuation marks disrupt the grammatical structure of the sentence.

3

Evaluate the choices to find the one that uses a symmetrical pair of punctuation marks to isolate the nonessential clause.

The choice using a comma at both the beginning and the end of the clause is correct, while all other options use asymmetrical patterns.

Only the matching pair of commas conforms to the conventions of Standard English.

Key Concept

Punctuation of Nonessential Elements

Question 953Question →

If $\frac{2}{5}(3x - 4) - \frac{1}{3}(2x + 5) = \frac{1}{5}x + \frac{11}{15}$ , what is the value of $2x - 7$ ?

Show answer & explanation

Answer: 17

Answer

The correct answer is 17.

To find the value of

2x - 7

, first solve the linear equation for

x

. Distributing the coefficients on the left side of the equation gives

\frac{6}{5}x - \frac{8}{5} - \frac{2}{3}x - \frac{5}{3} = \frac{1}{5}x + \frac{11}{15}

. Combining the variable terms and constants on the left side results in

\frac{8}{15}x - \frac{49}{15} = \frac{3}{15}x + \frac{11}{15}

. Subtracting

\frac{3}{15}x

and adding

\frac{49}{15}

to both sides yields

\frac{5}{15}x = \frac{60}{15}

, which simplifies to

\frac{1}{3}x = 4

, or

x = 12

. Finally, substituting 12 into the expression

2x - 7

gives

2(12) - 7 = 17

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses.

\frac{6}{5}x - \frac{8}{5} - \frac{2}{3}x - \frac{5}{3} = \frac{1}{5}x + \frac{11}{15}

To eliminate parentheses and allow grouping of like terms.

2

Combine the variable terms and the constant terms on the left side using a common denominator of 15.

\frac{8}{15}x - \frac{49}{15} = \frac{3}{15}x + \frac{11}{15}

To simplify the linear equation into a standard two-sided form.

3

Subtract the variable term from the right side and add the constant term from the left side.

\frac{5}{15}x = \frac{60}{15}

, which simplifies to

x = 12

To isolate the variable

x

on one side of the equation.

4

Evaluate the expression

2x - 7

using the value of

x

.

2(12) - 7 = 17

To solve for the final requested quantity.

Key Concept

Linear Equations in One Variable

Question 954Question →

Using a high-resolution scanning electron microscope to examine the fossilized leaves, _______ Which choice completes the text so that it conforms to the conventions of Standard English?

the paleontologists identified tiny structures indicative of ancient symbiotic fungi.

tiny structures indicative of ancient symbiotic fungi were identified by the paleontologists.

the microscope allowed the paleontologists to identify tiny structures indicative of ancient symbiotic fungi.

the analysis of the fossilized leaves by the paleontologists revealed tiny structures indicative of ancient symbiotic fungi.

Show answer & explanation

Answer: the paleontologists identified tiny structures indicative of ancient symbiotic fungi.

Answer

the paleontologists identified tiny structures indicative of ancient symbiotic fungi.

The correct option begins with the noun phrase 'the paleontologists,' which is the logical subject of the introductory modifying phrase 'Using a high-resolution scanning electron microscope to examine the fossilized leaves.' This placement ensures that the modifier correctly and clearly describes the entity performing the action.

Step-by-Step Solution

1

Identify the introductory modifying phrase and its logical subject.

The phrase is 'Using a high-resolution scanning electron microscope to examine the fossilized leaves'. The logical subject who performs the action of 'using' the microscope to 'examine' the leaves must be the researchers, or the paleontologists.

An introductory modifying phrase must logically describe the noun or pronoun that immediately follows it.

2

Evaluate the choices to find which one places the logical subject immediately after the comma.

Only the correct option begins with 'the paleontologists', establishing a logical and grammatically correct modifier-subject relationship.

Placing any other noun (such as 'tiny structures', 'the microscope', or 'the analysis') immediately after the comma creates a dangling or misplaced modifier, as those nouns cannot logically perform the action of using a microscope.

Key Concept

Subject-Modifier Agreement

Question 955Question →

Complete the text by filling in the blank with the correct phrase from the options in parentheses that conforms to the conventions of Standard English.

Fill in the blanks below

A pioneer of modern dance who rejected the rigid constraints of classical ballet, sought to express raw human emotion through natural, fluid movements. (Isadora Duncan / the choreography of Isadora Duncan)

Show answer & explanation

Answer

Isadora Duncan

The introductory modifier 'A pioneer of modern dance who rejected the rigid constraints of classical ballet' is an appositive phrase that must describe a person. The name 'Isadora Duncan' refers to the dancer, making it the grammatically correct subject of the main clause.

Step-by-Step Solution

1

Identify the introductory modifying phrase and determine the noun it logically describes.

The introductory modifier is 'A pioneer of modern dance who rejected the rigid constraints of classical ballet.' This appositive phrase must logically modify a person.

An introductory appositive modifier must directly describe the subject of the main clause that immediately follows it.

2

Compare the subject options to see which one represents the logical subject of the modifier.

The name 'Isadora Duncan' refers to the actual dancer (a person), while 'the choreography of Isadora Duncan' refers to her artistic works.

Only Isadora Duncan herself can be described as a 'pioneer' who 'rejected' ballet constraints, so she must be the subject of the sentence.

Key Concept

Subject-Modifier Agreement

Estimated Time:1m 0s

Question 956Question →

The table below shows some values of the linear function $f$ .

$x$	$f(x)$
$2$	$11$
$4$	$17$
$6$	$23$

What is the $y$ -coordinate of the $y$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5, representing the y-coordinate of the y-intercept of the graph of f.

To find the

y

-intercept of the linear function, we first determine its slope using the points

(2, 11)

and

(4, 17)

from the table. The slope

m

is

\frac{17 - 11}{4 - 2} = \frac{6}{2} = 3

. Next, we use the slope-intercept equation

f(x) = mx + b

. Substituting

m = 3

and the point

(2, 11)

gives

11 = 3(2) + b

, which simplifies to

11 = 6 + b

. Solving for

b

yields

5

. Therefore, the

y

-coordinate of the

y

-intercept of the graph of

f

is

5

.

Step-by-Step Solution

1

Calculate the slope (

m

) of the linear function using two coordinate pairs from the table.

The slope is

m = 3

.

A linear function has a constant slope, which can be found using the formula

m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

.

2

Substitute the slope and one of the points into the slope-intercept equation

f(x) = mx + b

to solve for the

y

-intercept

b

.

The

y

-intercept

b

is

5

.

The

y

-coordinate of the

y

-intercept of the graph of

y = f(x)

is the value of

b

in the equation

f(x) = mx + b

.

Key Concept

Finding the y-intercept of a linear function from a table of values.

Estimated Time:45s

Question 957Question →

Written in 1937 during a field trip to Haiti, _______

Which choice completes the text so that it conforms to the conventions of Standard English?

the novel *Their Eyes Were Watching God* was completed by author Zora Neale Hurston in only seven weeks.

author Zora Neale Hurston completed the novel *Their Eyes Were Watching God* in only seven weeks.

it took author Zora Neale Hurston only seven weeks to complete the novel *Their Eyes Were Watching God*.

only seven weeks were needed by author Zora Neale Hurston to complete the novel *Their Eyes Were Watching God*.

Show answer & explanation

Answer: the novel *Their Eyes Were Watching God* was completed by author Zora Neale Hurston in only seven weeks.

Answer

The correct answer completes the sentence with the novel as the subject, stating 'the novel *Their Eyes Were Watching God* was completed by author Zora Neale Hurston in only seven weeks.'

The correct option is the one that begins with the novel itself: 'the novel *Their Eyes Were Watching God* was completed by author Zora Neale Hurston in only seven weeks.' The introductory participial phrase 'Written in 1937 during a field trip to Haiti' must modify the subject of the main clause that follows it. Since the novel is what was written in 1937, the novel must be the subject of the main clause. This option correctly structures the sentence so that 'the novel' immediately follows the modifier.

Step-by-Step Solution

1

Identify the introductory modifying phrase and determine what it logically describes.

The introductory phrase 'Written in 1937 during a field trip to Haiti' describes a written work, which is the novel *Their Eyes Were Watching God*.

An introductory modifier must immediately precede and describe the logical subject of the sentence to avoid a dangling modifier.

2

Evaluate the options to find the one where the logical subject immediately follows the comma.

Only the option beginning with 'the novel *Their Eyes Were Watching God*' correctly places the logical subject immediately after the modifier.

This arrangement ensures subject-modifier agreement, whereas the other options place 'author Zora Neale Hurston', 'it', or 'only seven weeks' as the subject, resulting in dangling modifiers.

Key Concept

Subject-Modifier Agreement

Question 958Question →

In the $xy$ -plane, line $l_1$ passes through the origin and has a positive slope. Line $l_2$ is perpendicular to $l_1$ and intersects the $x$ -axis at $(d, 0)$ , where $d > 0$ . The two lines intersect at the point $(a, b)$ in the first quadrant. If the ratio of $a$ to $b$ is $3$ to $4$ , and the area of the triangle bounded by $l_1$ , $l_2$ , and the $x$ -axis is $A$ , which of the following expressions represents $d$ in terms of $A$ ?

\frac{5\sqrt{6A}}{8}

\frac{25A}{6}

\frac{5\sqrt{6A}}{6}

\frac{10\sqrt{6A}}{9}

Show answer & explanation

Answer: $\frac{5\sqrt{6A}}{6}$

Answer

The expression

\frac{5\sqrt{6A}}{6}

represents

d

in terms of

A

.

The correct expression is

\frac{5\sqrt{6A}}{6}

. The slope of line

l_1

passing through the origin and

(a,b)

is given by

m_1 = \frac{b}{a}

. Since the ratio of

a

to

b

is

3

to

4

, we have

\frac{a}{b} = \frac{3}{4}

, which implies the slope

m_1 = \frac{4}{3}

. Because line

l_2

is perpendicular to

l_1

, its slope is

m_2 = -\frac{3}{4}

. The equation of

l_2

passing through

(d,0)

is

y = -\frac{3}{4}(x - d)

. Solving the system of equations gives the intersection coordinates

a = \frac{9}{25}d

and

b = \frac{12}{25}d

. The area of the triangle bounded by the two lines and the

x

-axis is

A = \frac{1}{2} \times d \times b = \frac{6}{25}d^2

. Solving for

d

yields

d = \sqrt{\frac{25A}{6}} = \frac{5\sqrt{6A}}{6}

.

Step-by-Step Solution

1

Determine the slope and equation of line

l_1

.

Slope

m_1 = \frac{4}{3}

, and the equation of the line is

y = \frac{4}{3}x

.

Since

l_1

passes through

(0,0)

and the point

(a,b)

in the first quadrant, its slope is

m_1 = \frac{b}{a}

. Since the ratio of

a

to

b

is

3

to

4

, we have

\frac{a}{b} = \frac{3}{4} \implies \frac{b}{a} = \frac{4}{3}

.

2

Determine the equation of line

l_2

.

The equation of the line is

y = -\frac{3}{4}(x - d)

.

Line

l_2

is perpendicular to

l_1

, so its slope is the negative reciprocal of

m_1

, which is

m_2 = -\frac{3}{4}

. Using the point-slope form with the

x

-intercept

(d,0)

, the equation is

y - 0 = -\frac{3}{4}(x - d)

.

3

Find the coordinates of the intersection point

(a,b)

in terms of

d

.

a = \frac{9}{25}d

and

b = \frac{12}{25}d

.

Set the two equations equal to find the

x

-coordinate of the intersection:

\frac{4}{3}x = -\frac{3}{4}(x - d) \implies \frac{4}{3}x = -\frac{3}{4}x + \frac{3}{4}d \implies \frac{25}{12}x = \frac{3}{4}d \implies x = \frac{9}{25}d

. Substituting this back into the equation of

l_1

yields the

y

-coordinate:

y = \frac{4}{3}\left(\frac{9}{25}d\right) = \frac{12}{25}d

.

4

Express the area of the triangle in terms of

d

.

A = \frac{6}{25}d^2

.

The base of the triangle along the

x

-axis is

d

(from

x=0

to

x=d

). The height of the triangle is the

y

-coordinate of the intersection point,

b = \frac{12}{25}d

. The area of the triangle is given by

A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}d\left(\frac{12}{25}d\right) = \frac{6}{25}d^2

.

5

Solve for

d

in terms of

A

.

d = \frac{5\sqrt{6A}}{6}

.

Isolate

d^2

to get

d^2 = \frac{25A}{6}

. Taking the square root of both sides gives

d = \sqrt{\frac{25A}{6}} = \frac{5\sqrt{A}}{\sqrt{6}} = \frac{5\sqrt{6A}}{6}

.

Key Concept

Using the properties of perpendicular lines, setting up equations from coordinate parameters, and applying geometric formulas to relate parameters in linear systems.

Alternative Method

Using the geometric mean theorem (altitude rule) in a right triangle, the altitude

b

divides the hypotenuse

d

into segments

a

and

d-a

, such that

b^2 = a(d-a)

. Since

b = \frac{4}{3}a

, we can substitute this to find

a = \frac{9}{25}d

and

b = \frac{12}{25}d

directly without finding the line equations. Then,

A = \frac{1}{2} d b = \frac{6}{25}d^2

, which solves to

d = \frac{5\sqrt{6A}}{6}

.

Estimated Time:3m 0s

Question 959Question →

In the $xy$ -plane, the graph of the linear function $f$ is perpendicular to the line with equation $3x + 4y = 48$ . The graph of $f$ intersects the $x$ -axis at the point $(p, 0)$ and the $y$ -axis at the point $(0, q)$ , where $q > 0$ . If the distance between the points $(p, 0)$ and $(0, q)$ is $15$ , what is the value of $q$ ?

Show answer & explanation

Answer: 12

Answer

12

First, find the slope of the given line by rewriting

3x + 4y = 48

in slope-intercept form:

y = -\frac{3}{4}x + 12

. The slope is

-\frac{3}{4}

. The graph of the linear function

f

is perpendicular to this line, so its slope is the negative reciprocal,

\frac{4}{3}

. With a

y

-intercept of

(0, q)

, the equation of

f

is

y = \frac{4}{3}x + q

. Setting

y = 0

gives the

x

-intercept

(p, 0) = (-\frac{3}{4}q, 0)

. The distance between these intercepts is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

(since

q > 0

). Given that the distance is

15

, we solve

\frac{5}{4}q = 15

to find

q = 12

.

Step-by-Step Solution

1

Find the slope of the line

3x + 4y = 48

.

The slope of the line is

-\frac{3}{4}

.

To find the slope of the perpendicular line

f

, we first need the slope of the given line. Rewriting

3x + 4y = 48

in slope-intercept form gives

y = -\frac{3}{4}x + 12

.

2

Determine the slope of

f

.

The slope of

f

is

\frac{4}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Find the

x

-intercept of

f

in terms of

q

.

The

x

-intercept of

f

is

(-\frac{3}{4}q, 0)

.

Since the

y

-intercept of

f

is

(0, q)

, the equation of

f

is

y = \frac{4}{3}x + q

. Setting

y = 0

gives

0 = \frac{4}{3}x + q

, which simplifies to

x = -\frac{3}{4}q

.

4

Use the distance formula between the intercepts to solve for

q

.

The value of

q

is

12

.

The distance between

(-\frac{3}{4}q, 0)

and

(0, q)

is

\sqrt{(-\frac{3}{4}q)^2 + q^2} = \sqrt{\frac{25}{16}q^2} = \frac{5}{4}q

since

q > 0

. Setting this distance to

15

gives

\frac{5}{4}q = 15

, which yields

q = 12

.

Key Concept

Properties of perpendicular lines, finding intercepts, and utilizing the distance formula in coordinate geometry.

Question 960Question →

An online store charges a flat shipping fee of $8.50$ dollars for orders under $100$ dollars. A customer purchases $5$ identical shirts and uses a coupon for $12.00$ dollars off the total price of the shirts. If the total charge for the order, including shipping, is $86.50$ dollars, what is the price, in dollars, of one shirt?

13.20

15.60

18.00

19.70

Show answer & explanation

Answer: $18.00$

Answer

The price of one shirt is

18.00

dollars.

To find the price of one shirt, we set up a linear equation. Let

s

represent the price of one shirt. The cost of

5

shirts after applying the

12.00

dollar coupon is

5s - 12.00

. Since this amount is under

100

dollars, the flat shipping fee of

8.50

dollars is added to the total. This gives the equation

5s - 12.00 + 8.50 = 86.50

. Combining the constants on the left side results in

5s - 3.50 = 86.50

. Adding

3.50

to both sides yields

5s = 90.00

. Dividing both sides by

5

gives

s = 18.00

. Thus, the price of one shirt is

18.00

dollars.

Step-by-Step Solution

1

Represent the cost of the shirts using a variable.

Let

s

represent the price of one shirt. The cost of

5

shirts with the

12.00

dollar coupon discount is

5s - 12.00

dollars.

This establishes the variable for the unknown value we need to find.

2

Set up the full linear equation by including the shipping fee and total charge.

Since

5s - 12.00

is under

100

dollars (which we verify at the end), a flat shipping fee of

8.50

dollars applies. The equation is

5s - 12.00 + 8.50 = 86.50

.

This relates all parts of the word problem into a single solvable mathematical statement.

3

Simplify the equation by combining the constant values on the left side.

5s - 3.50 = 86.50

Combining like terms simplifies the algebraic expression.

4

Isolate the variable term by adding

3.50

to both sides.

5s = 90.00

This moves all constant terms to one side of the equation.

5

Solve for the variable by dividing both sides of the equation by

5

.

s = 18.00

Dividing by the coefficient of the variable isolates

s

to find the final value.

Key Concept

Linear Equations in One Variable

Answer

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