All practice questions

Question 101Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\sin(A) - \cos(A) = \frac{1}{5}$ , what is the value of $\sin(A)\sin(B)$ ?

\frac{13}{25}

\frac{16}{25}

\frac{12}{25}

\frac{24}{25}

Show answer & explanation

Answer: $\frac{12}{25}$

Answer

\frac{12}{25}

By the co-function identity for complementary angles in a right triangle,

\sin(B) = \cos(A)

, which means the expression

\sin(A)\sin(B)

is equivalent to

\sin(A)\cos(A)

. Squaring the equation

\sin(A) - \cos(A) = \frac{1}{5}

gives

\sin^2(A) - 2\sin(A)\cos(A) + \cos^2(A) = \frac{1}{25}

. Substituting the Pythagorean identity

\sin^2(A) + \cos^2(A) = 1

simplifies this to

1 - 2\sin(A)\cos(A) = \frac{1}{25}

. Solving for

\sin(A)\cos(A)

yields

2\sin(A)\cos(A) = \frac{24}{25}

, or

\sin(A)\cos(A) = \frac{12}{25}

.

Step-by-Step Solution

1

Express the target expression in terms of a single angle using complementary relationships.

Since angle

C = 90^\circ

, angles

A

and

B

are complementary, so

\sin(B) = \cos(A)

. Thus,

\sin(A)\sin(B) = \sin(A)\cos(A)

.

In any right triangle, the sine of one acute angle equals the cosine of the other acute angle.

2

Square both sides of the given equation

\sin(A) - \cos(A) = \frac{1}{5}

.

(\sin(A) - \cos(A))^2 = \left(\frac{1}{5}\right)^2 \implies \sin^2(A) - 2\sin(A)\cos(A) + \cos^2(A) = \frac{1}{25}

.

Squaring the difference allows us to introduce the product term

\sin(A)\cos(A)

and the squared terms.

3

Apply the Pythagorean identity to simplify the equation.

Since

\sin^2(A) + \cos^2(A) = 1

, the equation becomes

1 - 2\sin(A)\cos(A) = \frac{1}{25}

.

The sum of the squares of sine and cosine of the same angle is always equal to 1.

4

Solve for the product

\sin(A)\cos(A)

.

2\sin(A)\cos(A) = 1 - \frac{1}{25} = \frac{24}{25} \implies \sin(A)\cos(A) = \frac{12}{25}

.

Subtracting

1/25

from

1

and dividing the resulting fraction by

2

isolates the target product.

Key Concept

Pythagorean identity and co-function relationships in right triangles

Question 102Question →

The amount of a radioactive isotope remaining after $t$ days is modeled by the function $A(t) = A_0(0.64)^{\frac{t}{2}}$ , where $A_0$ is the initial amount of the isotope. If the function is rewritten in the form $A(t) = A_0(1 - r)^t$ , where $r$ is the daily decay rate, what is the value of $r$ ?

0.18

0.20

0.36

0.59

Show answer & explanation

Answer: 0.20

Answer

The daily decay rate

r

is

0.20

.

The correct value of

r

is

0.20

. By rewriting the given expression using exponent rules, we obtain

(0.64)^{t/2} = ((0.64)^{1/2})^t = (0.8)^t

. Setting the base equal to the target decay rate form gives

1 - r = 0.8

, which simplifies to

r = 0.20

.

Step-by-Step Solution

1

Apply the power of a power exponent rule,

(x^a)^b = x^{ab}

, to rewrite the exponential term.

A(t) = A_0\left(0.64^{1/2}\right)^t

This isolates the variable

t

as the exponent, allowing us to find the daily decay factor.

2

Evaluate the base

0.64^{1/2}

.

\sqrt{0.64} = 0.8

, so the expression becomes

A(t) = A_0(0.8)^t

.

Calculating the square root of

0.64

gives the daily decay factor of

0.8

.

3

Equate the daily decay factor

0.8

to the target form

1 - r

and solve for

r

.

1 - r = 0.8 \implies r = 0.20

Solving the equation gives the value of the daily decay rate.

Key Concept

Rewriting exponential equations by manipulating bases and exponents

Alternative Method

We can solve for

r

by choosing a specific value for

t

, such as

t = 2

. After 2 days, the remaining amount is

A(2) = A_0(0.64)^1 = 0.64A_0

. Using the target equation form, the remaining amount after 2 days is

A_0(1 - r)^2

. Setting the two expressions equal to each other gives

A_0(1 - r)^2 = 0.64A_0

. Dividing both sides by

A_0

gives

(1 - r)^2 = 0.64

. Taking the square root of both sides gives

1 - r = 0.8

, which yields

r = 0.20

.

Estimated Time:1m 30s

Question 103Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . The acute angles $A$ and $C$ satisfy $\sin(A) = \frac{k}{3}$ and $\cos(C) = \frac{4}{k+4}$ for some positive constant $k$ . What is the value of $k$ ?

6

8

2

12

Show answer & explanation

Answer: 2

Answer

2

In right triangle

ABC

, the measure of angle

B

is

90^\circ

, which means the acute angles

A

and

C

are complementary. Therefore, the co-function identity states that

\sin(A) = \cos(C)

. Equating the given expressions yields

\frac{k}{3} = \frac{4}{k+4}

. Cross-multiplying results in

k(k+4) = 12

, which simplifies to

k^2 + 4k - 12 = 0

. Factoring this equation gives

(k+6)(k-2) = 0

. This quadratic has solutions

k = -6

and

k = 2

. Since

k

must be positive, the value of

k

is

2

.

Step-by-Step Solution

1

Identify the relationship between the acute angles in a right triangle.

Since the measure of angle

B

is

90^\circ

, the sum of angles

A

and

C

is

90^\circ

, which means they are complementary angles. By the co-function identity,

\sin(A) = \cos(C)

.

In a right triangle, the sine of one acute angle equals the cosine of the other acute angle.

2

Equate the expressions for

\sin(A)

and

\cos(C)

and set up the equation for

k

.

\frac{k}{3} = \frac{4}{k+4}

This sets up the algebraic relation to solve for

k

using the given trigonometric expressions.

3

Solve the algebraic equation for

k

.

Cross-multiplying gives

k(k+4) = 12

, which expands to

k^2 + 4k = 12

. Subtracting

12

from both sides results in the quadratic equation

k^2 + 4k - 12 = 0

. Factoring this equation yields

(k+6)(k-2) = 0

.

Cross-multiplication eliminates the denominators, converting the equation to a quadratic form that can be solved by factoring.

4

Determine the valid positive value for

k

.

The solutions to the equation

(k+6)(k-2) = 0

are

k = -6

and

k = 2

. Since

k

must be a positive constant, we select

k = 2

.

The problem specifies that

k

is a positive constant, so the negative solution is discarded.

Key Concept

Co-function identity relating sine and cosine of complementary angles

Estimated Time:1m 30s

Question 104Question →

If the expression $(x + 4)^2 - (x - 4)^2$ is equivalent to $kx$ for all values of $x$ , where $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 16

Answer

The value of the constant

k

is

16

.

To find the value of

k

, the expression

(x + 4)^2 - (x - 4)^2

is simplified by expanding both binomials. The first binomial expands to

x^2 + 8x + 16

, and the second binomial expands to

x^2 - 8x + 16

. Subtracting the second expression from the first requires distributing the negative sign across all terms:

(x^2 + 8x + 16) - (x^2 - 8x + 16) = x^2 - x^2 + 8x - (-8x) + 16 - 16 = 16x

. Equating

16x

to

kx

yields

k = 16

.

Step-by-Step Solution

1

Expand the first squared term,

(x + 4)^2

x^2 + 8x + 16

Applying the binomial expansion formula

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

where

a = x

and

b = 4

.

2

Expand the second squared term,

(x - 4)^2

x^2 - 8x + 16

Applying the binomial expansion formula

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

where

a = x

and

b = 4

.

3

Subtract the second expanded expression from the first and simplify

16x

Group like terms and distribute the negative sign:

(x^2 + 8x + 16) - (x^2 - 8x + 16) = x^2 - x^2 + 8x + 8x + 16 - 16 = 16x

.

4

Equate the simplified expression to

kx

to determine the constant

k

k = 16

Since

16x = kx

for all values of

x

, the coefficients must be equal.

Key Concept

Equivalent Algebraic Expressions

Question 105Question →

In the study of historical sociology, researchers often examine how structural changes in communication technologies influence social movements. While early theories suggested that the printing press directly caused the Protestant Reformation, contemporary scholars argue that the technology’s impact was actually ________ by pre-existing trade networks and literacy rates, which dictated where and how quickly new ideas could spread.

Which choice completes the text with the most logical and precise word or phrase?

precipitated

mediated

nullified

overlooked

Show answer & explanation

Answer: mediated

Answer

mediated

The correct answer is 'mediated'. In this context, 'mediated' means influenced, channeled, or moderated by an intervening factor. The passage contrasts the early theory of direct causation with the contemporary view that the printing press's impact was shaped or channeled by pre-existing trade networks and literacy rates.

Step-by-Step Solution

1

Analyze the context of the sentence containing the blank.

The sentence sets up a contrast between an early theory ('printing press directly caused the Protestant Reformation') and contemporary views ('was actually ________ by pre-existing trade networks').

Understanding the contrast helps determine that the blank must describe how the printing press's impact was modified or channeled by other factors, rather than being direct.

2

Evaluate the choices to find the word that means influenced, channeled, or moderated by intervening factors.

The word 'mediated' fits this definition perfectly, as the trade networks and literacy rates acted as intervening channels for the technology's impact.

Selecting the option that matches the contextual meaning of moderation or channeling by secondary factors.

Key Concept

Words in Context: High-Utility Vocabulary

Estimated Time:1m 0s

Question 106Question →

For decades, agricultural scientists assumed that soil quality was determined solely by chemical composition. However, recent research highlights the role of soil microbiomes, demonstrating that diverse microbial communities are crucial to plant health. Consequently, focusing exclusively on chemical inputs represents a ___________ approach that overlooks the dynamic, biological systems essential for sustainable crop yields.

Which choice completes the text with the most logical and precise word or phrase?

rigorous

myopic

holistic

innovative

Show answer & explanation

Answer: myopic

Answer

The correct answer is the word describing a narrow or short-sighted approach, which is 'myopic'.

The correct option is the word that describes a short-sighted or narrow perspective. In the context of the passage, focusing solely on chemical inputs and neglecting the crucial biological aspects of soil microbiomes is described as a narrow approach that overlooks essential factors. The word 'myopic' precisely conveys this meaning of being short-sighted or lacking foresight.

Step-by-Step Solution

1

Analyze the context of the sentence containing the blank.

The sentence indicates that focusing 'exclusively on chemical inputs' results in an approach that 'overlooks the dynamic, biological systems' of the soil.

This shows that the blank requires a word describing an approach that is narrow, short-sighted, or limited in its view.

2

Evaluate the choices to find the word that best conveys a short-sighted or narrow perspective.

The word 'myopic' means short-sighted or lacking foresight, which precisely describes focusing on one narrow aspect while ignoring the broader biological reality. Other choices do not match this meaning.

This identifies the correct choice based on contextual meaning.

Key Concept

Contextual vocabulary usage requires identifying logical transitions and connotations in a text to select the most precise word.

Question 107Question →

Sociologist Dr. Liam O'Connor and colleagues investigated the social impact of urban "pocket parks"—small public green spaces built on single vacant lots. O'Connor hypothesized that pocket parks with active community-led stewardship, such as neighborhood-organized planting groups, foster stronger local social cohesion than pocket parks maintained exclusively by municipal government workers. To test this hypothesis, the researchers surveyed residents living within a two-block radius of various pocket parks.

Which choice, if true, would most directly support O'Connor and colleagues' hypothesis?

Pocket parks with community-led stewardship featured a wider variety of plant species and higher biodiversity than parks maintained by municipal workers.

Surveyed residents near all pocket parks reported visiting green spaces more frequently than they did before the parks were built.

Residents living near community-stewardship parks reported knowing their neighbors by name and participating in local cooperative efforts at a significantly higher rate than residents near municipally maintained parks.

Residents living near municipally maintained pocket parks expressed a strong desire to take over maintenance of those parks themselves.

Show answer & explanation

Answer: Residents living near community-stewardship parks reported knowing their neighbors by name and participating in local cooperative efforts at a significantly higher rate than residents near municipally maintained parks.

Answer

The option stating that residents near community-stewardship parks reported knowing their neighbors by name and participating in local cooperative efforts at a higher rate.

The correct choice directly supports the hypothesis by showing a clear, positive correlation between community-led stewardship and indicators of social cohesion (knowing neighbors by name and cooperating locally) compared to municipally maintained parks.

Step-by-Step Solution

1

Identify the researchers' hypothesis in the passage.

The hypothesis is that pocket parks with community-led stewardship foster stronger local social cohesion than parks maintained exclusively by municipal workers.

Understanding the precise claim is necessary to determine what evidence would support it.

2

Evaluate the choices to find one that compares the two types of parks in terms of social cohesion.

The correct option shows a clear difference in neighborhood ties (knowing neighbors, cooperative efforts) in favor of community-led stewardship parks.

Supporting evidence must directly connect community stewardship to increased social cohesion.

3

Eliminate choices that focus on unrelated outcomes or fail to compare the two types of parks.

Options about biodiversity, general park visits, or future desires are eliminated because they do not support the specific comparative claim.

Distractors often present true statements or related concepts that do not address the specific hypothesis.

Key Concept

Identifying textual evidence that directly supports a specific research hypothesis.

Question 108Question →

Lines $L_1$ and $L_2$ are parallel. Points $P$ and $Q$ lie on line $L_1$ , and points $R$ and $S$ lie on line $L_2$ . Segment $PS$ and segment $QR$ intersect at point $T$ . If the measure of angle $TPQ$ is $38^\circ$ and the measure of angle $PTQ$ is $105^\circ$ , what is the measure, in degrees, of angle $TRS$ ?

Show answer & explanation

Answer: 37

Answer

37

To find the measure of angle

TRS

, we first find the measure of angle

TQP

using the property that the sum of interior angles in triangle

TPQ

is

180^\circ

. This gives

\angle TQP = 180^\circ - 38^\circ - 105^\circ = 37^\circ

. Since lines

L_1

and

L_2

are parallel, the transversal line

QR

creates alternate interior angles that are equal in measure. Therefore, angle

TRS

is equal to angle

TQP

, which is

37^\circ

.

Step-by-Step Solution

1

Calculate the measure of angle

TQP

in triangle

TPQ

.

\angle TQP = 37^\circ

The sum of the angles in a triangle is

180^\circ

. Therefore,

\angle TQP = 180^\circ - \angle TPQ - \angle PTQ = 180^\circ - 38^\circ - 105^\circ = 37^\circ

.

2

Relate angle

TQP

to angle

TRS

using parallel line properties.

\angle TRS = 37^\circ

Since line

L_1

is parallel to line

L_2

, the transversal line

QR

creates equal alternate interior angles. Thus,

\angle TRS = \angle TQP = 37^\circ

.

Key Concept

Alternate interior angles and triangle angle sum theorem

Question 109Question →

In a 2018 study of traditional Andean music, ethnomusicologist Sofia Morales analyzed the *caja*, a small hand-held drum played during agricultural festivals. Many scholars have argued that the instrument's primary function is strictly rhythmic, acting as a simple metronome for communal dances. However, Morales observed that players adjust their striking technique to produce specific resonant overtones that mimic the calls of highland birds during planting seasons. This subtle acoustic imitation does not merely accompany the ritual; rather, it actively integrates the natural environment into the performance. Thus, the *caja* serves a complex symbolic role, mediating the community's relationship with their ecological surroundings.

Which choice best describes the function of the underlined sentence in the text as a whole?

It introduces a specific observation that complicates a common assumption about the primary function of the *caja*.

It states the final conclusion of the passage regarding the symbolic role of the *caja* in Andean communities.

It outlines the primary thesis that the passage as a whole attempts to disprove.

It provides detailed evidence that supports the claim that the drum acts solely as a rhythmic metronome.

Show answer & explanation

Answer: It introduces a specific observation that complicates a common assumption about the primary function of the *caja*.

Answer

The underlined sentence introduces a specific observation that complicates a common assumption about the primary function of the *caja*.

The correct answer identifies that the underlined sentence introduces a specific observation (players modifying their technique to mimic bird calls) that complicates the previous claim (that the drum serves a strictly rhythmic function). This is marked by the contrast transition word 'However' and directly transitions the reader from the traditional view to Morales's new interpretation.

Step-by-Step Solution

1

Analyze the context before the underlined sentence.

The text states that many scholars view the *caja* as having a strictly rhythmic function, acting as a simple metronome.

This establishes the baseline consensus that the author is responding to.

2

Analyze the underlined sentence itself.

The underlined sentence begins with 'However' and notes that players adjust their technique to mimic bird calls.

This contrasts with the previous claim and presents a new finding about the drum's actual musical use.

3

Analyze the sentences after the underlined sentence.

The text explains that this imitation integrates the natural environment and concludes that the drum serves a complex symbolic role.

This demonstrates how the finding in the underlined sentence serves as a pivot or foundation for the passage's final conclusion.

4

Match the findings with the correct option.

The correct option describes this function as introducing an observation that complicates the common assumption.

It accurately represents the part-to-whole relationship of the sentence within the paragraph's structure.

Key Concept

Part-to-Whole Rhetorical Function

Estimated Time:1m 15s

Question 110Question →

In marine ecosystems, the dinoflagellate *Noctiluca scintillans* exhibits bioluminescence when physically disturbed by predators. Researchers hypothesized that this bioluminescence acts as a 'burglar alarm,' illuminating the predator and attracting larger secondary predators that consume the primary predator. To test this, researchers placed the copepod *Temora longicornis* (a primary predator of *N. scintillans*) and three-spined stickleback fish (a secondary predator) in tanks containing *N. scintillans*. In dark conditions, stickleback predation on copepods was significantly higher in tanks with bioluminescent *N. scintillans* than in those with a non-bioluminescent mutant strain, suggesting that ______

Which choice most logically completes the text?

bioluminescent organisms in marine environments are generally less vulnerable to predation than non-bioluminescent organisms.

the bioluminescence of *N. scintillans* aids stickleback fish in locating copepods under low-light conditions.

copepods are more active and easier to catch when they are feeding on bioluminescent prey than on non-bioluminescent prey.

stickleback fish prefer to consume *N. scintillans* directly when copepods are not present in the tanks.

Show answer & explanation

Answer: the bioluminescence of *N. scintillans* aids stickleback fish in locating copepods under low-light conditions.

Answer

The bioluminescence of *N. scintillans* aids stickleback fish in locating copepods under low-light conditions.

The correct option is supported because the experiment showed that sticklebacks successfully preyed on more copepods in dark conditions only when the dinoflagellates produced bioluminescence. This directly supports the idea that the bioluminescence acts as a beacon, helping the fish locate the copepods in the dark.

Step-by-Step Solution

1

Identify the main premise and hypothesis in the passage.

The passage states that *N. scintillans* bioluminesces when disturbed, and researchers hypothesized this attracts secondary predators (stickleback fish) to consume primary predators (copepods) by illuminating them.

Understanding the setup helps determine what the experimental results should show.

2

Analyze the experimental results.

In dark conditions, stickleback predation on copepods was significantly higher in tanks with bioluminescent *N. scintillans* than in those with a non-bioluminescent mutant strain.

The difference in predation rate must be attributed to the only variable changed, which is the bioluminescence.

3

Draw a logical inference based on the results.

Since predation by sticklebacks (which are secondary predators) on copepods was higher when bioluminescence was present, the bioluminescence must have made the copepods more visible or easier to locate for the sticklebacks in the dark.

This connects the experimental result back to the hypothesis that the bioluminescence aids secondary predators.

Key Concept

Logical Inferences

Estimated Time:1m 30s

Question 111Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\cos(A) = k \sin(A)$ for some positive constant $k$ , which of the following expressions represents $\cos(B)$ in terms of $k$ ?

\frac{k}{\sqrt{1 + k^2}}

\frac{1}{\sqrt{1 - k^2}}

\frac{1}{\sqrt{1 + k^2}}

\sqrt{1 + k^2}

Show answer & explanation

Answer: $\frac{1}{\sqrt{1 + k^2}}$

Answer

The expression that represents

\cos(B)

is

\frac{1}{\sqrt{1 + k^2}}

.

In right triangle

ABC

with right angle

C

, the acute angles

A

and

B

are complementary (

A + B = 90^\circ

). By the co-function identity,

\cos(B) = \sin(A)

. Using the given relation

\cos(A) = k \sin(A)

and substituting it into the Pythagorean identity

\sin^2(A) + \cos^2(A) = 1

, we get

\sin^2(A) + (k \sin(A))^2 = 1 \implies \sin^2(A)(1 + k^2) = 1

. Solving for

\sin(A)

gives

\sin(A) = \frac{1}{\sqrt{1 + k^2}}

because

\sin(A)

must be positive for an acute angle. Since

\cos(B) = \sin(A)

, we conclude that the correct expression is

\frac{1}{\sqrt{1 + k^2}}

.

Step-by-Step Solution

1

Use the relationship between the acute angles in right triangle

ABC

.

Since angle

C

is

90^\circ

, angles

A

and

B

are complementary. Thus,

\cos(B) = \sin(A)

.

This allows us to convert the target term

\cos(B)

into

\sin(A)

, which can be related to the given equation.

2

Substitute the given relation

\cos(A) = k \sin(A)

into the Pythagorean identity.

Using

\sin^2(A) + \cos^2(A) = 1

, we substitute to get

\sin^2(A) + (k \sin(A))^2 = 1

, which simplifies to

\sin^2(A)(1 + k^2) = 1

.

This sets up a single equation containing only

\sin(A)

and the constant

k

.

3

Solve for

\sin(A)

and substitute back to find

\cos(B)

.

Solving for

\sin(A)

gives

\sin(A) = \frac{1}{\sqrt{1 + k^2}}

(since

\sin(A) > 0

for acute angle

A

). Since

\cos(B) = \sin(A)

, we have

\cos(B) = \frac{1}{\sqrt{1 + k^2}}

.

This yields the final value of

\cos(B)

in terms of

k

.

Key Concept

Applying complementary angle trigonometric identities (co-functions) and the Pythagorean identity in a right triangle.

Alternative Method

Alternatively, you can model this by setting up a right triangle. Since

\cos(A) = k \sin(A)

, we can divide both sides by

\sin(A)

to get

\cot(A) = k

, which means

\tan(A) = \frac{1}{k}

. In a right triangle

ABC

with right angle

C

,

\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{1}{k}

. Let

BC = 1

and

AC = k

. By the Pythagorean theorem, the hypotenuse

AB = \sqrt{1^2 + k^2} = \sqrt{1 + k^2}

. Then,

\cos(B) = \frac{\text{adjacent to } B}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{1}{\sqrt{1 + k^2}}

.

Estimated Time:2m 0s

Question 112Question →

The quadratic function $f$ is defined by $f(x) = -2x^2 + bx + c$ , where $b$ and $c$ are constants. In the $xy$ -plane, the vertex of the graph of $f$ has an $x$ -coordinate of $3$ . If $f(1) = 2$ , what is the $y$ -coordinate of the vertex of the graph of $f$ ?

Show answer & explanation

Answer: 10

Answer

10

The quadratic function in standard form is

f(x) = -2x^2 + bx + c

, which has a leading coefficient of

a = -2

. The vertex form of a quadratic function is

f(x) = a(x-h)^2 + k

, where

(h, k)

is the vertex of the parabola. Given that the

x

-coordinate of the vertex is

3

(so

h = 3

), we can write the function as

f(x) = -2(x-3)^2 + k

. Since the graph passes through the point

(1, 2)

, we substitute

x = 1

and

f(1) = 2

into the equation:

2 = -2(1-3)^2 + k

. Simplifying the expression gives

2 = -2(-2)^2 + k

, which becomes

2 = -2(4) + k

, or

2 = -8 + k

. Adding

8

to both sides yields

k = 10

. Thus, the

y

-coordinate of the vertex is

10

.

Step-by-Step Solution

1

Write the quadratic function in vertex form.

f(x) = -2(x-3)^2 + k

The standard form equation

f(x) = -2x^2 + bx + c

shows that the leading coefficient

a = -2

. The vertex form is

f(x) = a(x-h)^2 + k

, where

(h, k)

is the vertex. Since the

x

-coordinate of the vertex is

3

, we substitute

a = -2

and

h = 3

into the vertex form.

2

Substitute the point

(1, 2)

into the vertex form equation.

2 = -2(1-3)^2 + k

Since

f(1) = 2

, the graph passes through the point

(1, 2)

. Substituting these values allows us to solve for the unknown vertex

y

-coordinate,

k

.

3

Simplify the equation and solve for

k

.

k = 10

Simplifying the term

-2(1-3)^2

yields

-2(-2)^2 = -2(4) = -8

. This simplifies the equation to

2 = -8 + k

. Adding

8

to both sides gives

k = 10

.

Key Concept

Vertex form of a quadratic function

Question 113Question →

In the system of equations below, what is the value of $y$ ?

4x + 3y = 25

2x + 3y = 17

Show answer & explanation

Answer: 3

Answer

The value of

y

is

3

.

Subtracting the second equation from the first equation gives

(4x + 3y) - (2x + 3y) = 25 - 17

, which simplifies to

2x = 8

. Dividing by 2 gives

x = 4

. Substituting

x = 4

into the second equation gives

2(4) + 3y = 17

, which simplifies to

8 + 3y = 17

. Subtracting 8 from both sides gives

3y = 9

, and dividing by 3 gives

y = 3

.

Step-by-Step Solution

1

Subtract the second equation from the first equation to eliminate the

y

term.

2x = 8

Subtracting the equations eliminates

3y

since it is common to both equations, leaving a single variable equation.

2

Solve for

x

by dividing both sides of the equation by 2.

x = 4

Dividing isolates the variable

x

so we can find its numerical value.

3

Substitute

x = 4

into the second equation

2x + 3y = 17

and solve for

y

.

y = 3

Substituting the value of

x

leaves only the variable

y

, which can then be isolated and solved.

Key Concept

Solving systems of linear equations using elimination

Question 114Question →

In the field of volcanology, analyzing the microscopic structures within volcanic rock helps researchers reconstruct the history of past eruptions. Each of the olivine crystals retrieved from the ancient basaltic lava flows of Mount Etna ______ crucial data about the cooling rate and chemical composition of the magma chamber before the eruption occurred.

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

provides

provide

have provided

are providing

Show answer & explanation

Answer: provides

Answer

provides

The singular verb 'provides' agrees with the singular pronoun subject 'Each'. The intervening modifying phrase containing plural nouns does not alter the singular nature of the subject.

Step-by-Step Solution

1

Identify the true grammatical subject of the sentence.

The subject of the sentence is the singular indefinite pronoun 'Each'.

The prepositional phrase 'of the olivine crystals retrieved from the ancient basaltic lava flows of Mount Etna' contains plural nouns ('crystals', 'flows'), but it acts as a modifier and does not change the subject of the sentence.

2

Determine the correct verb form that agrees in number with the singular subject.

A singular verb is required.

A singular subject ('Each') must be paired with a singular verb form ('provides') to maintain grammatical agreement.

Key Concept

Subject-Verb Agreement

Question 115Question →

A circle in the $xy$ -plane is defined by the equation $(x - 7)^2 + (y + 2)^2 = 36$ . What is the diameter of this circle?

6

12

36

72

Show answer & explanation

Answer: 12

Answer

12

The standard equation of a circle is

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center of the circle and

r

is the radius. In the equation

(x - 7)^2 + (y + 2)^2 = 36

, the value of

r^2

is 36. Taking the square root gives the radius,

r = 6

. The diameter of a circle is twice its radius, so the diameter is

2(6) = 12

.

Step-by-Step Solution

1

Identify the value of the radius squared,

r^2

, from the given equation of the circle.

r^2 = 36

In the standard equation of a circle,

(x - h)^2 + (y - k)^2 = r^2

, the constant on the right side of the equation is equal to the square of the radius.

2

Determine the radius,

r

, by taking the square root of

r^2

.

r = \sqrt{36} = 6

Since the radius represents a distance, it must be the positive square root of the constant term.

3

Calculate the diameter of the circle by multiplying the radius by 2.

Diameter =

2 \times 6 = 12

The diameter of a circle is defined as twice its radius.

Key Concept

Identifying circle properties from its standard equation form

(x - h)^2 + (y - k)^2 = r^2

.

Question 116Question →

Two linear equations are defined as follows:

3x + 2y = 12

x - 2y = 4

If the ordered pair $(x, y)$ satisfies both equations, what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

Adding the two equations

3x + 2y = 12

and

x - 2y = 4

eliminates the

y

terms, resulting in

4x = 16

. Dividing both sides by 4 gives the value of

x

as 4.

Step-by-Step Solution

1

Add the two equations together to eliminate the

y

variable.

4x = 16

Since the coefficients of

y

are opposites (

2

and

-2

), adding the equations eliminates

y

directly.

2

Divide both sides of the equation by 4.

x = 4

To isolate the variable

x

.

Key Concept

Solving systems of linear equations using the elimination method.

Alternative Method

We can solve the second equation for

x

to get

x = 2y + 4

. Substituting this expression into the first equation gives

3(2y + 4) + 2y = 12

, which simplifies to

6y + 12 + 2y = 12

, or

8y = 0

, meaning

y = 0

. Substituting

y = 0

back into

x = 2y + 4

yields

x = 4

.

Estimated Time:45s

Question 117Question →

A student is preparing a presentation on the Lycurgus Cup and has taken the following notes:
* The Lycurgus Cup is a Roman glass chalice dating to the 4th century AD.
* It is made of dichroic glass, which displays different colors depending on how light hits it.
* When light is reflected off the front of the cup, the glass appears green.
* When light is transmitted through the glass from behind, the cup glows red.
* This color-changing property is caused by nanoparticles of gold and silver embedded in the glass.

The student wants to write a sentence that emphasizes the two specific colors the cup displays depending on the lighting direction.

Based on the notes, complete the sentence to achieve this goal.

Fill in the blanks below

Highlighting its unique properties, the Lycurgus Cup appears in reflected light but glows in transmitted light.

Show answer & explanation

Answer

Highlighting its unique properties, the Lycurgus Cup appears green in reflected light but glows red in transmitted light.

The completed sentence correctly identifies that the Lycurgus Cup appears green in reflected light and red in transmitted light. This directly satisfies the rhetorical goal of emphasizing the two specific colors the cup displays depending on the direction of the light, using the facts provided in the notes.

Step-by-Step Solution

1

Identify the rhetorical goal stated in the prompt.

The goal is to emphasize the two specific colors the cup displays depending on the lighting direction.

This determines which specific details from the bulleted notes must be synthesized in the sentence.

2

Locate the relevant details matching the goal from the notes.

The notes indicate the cup appears green when light is reflected off the front (reflected light) and red when light is transmitted from behind (transmitted light).

These details directly provide the specific colors and corresponding lighting directions needed.

3

Fill in the blanks to complete the sentence structure.

The first blank, associated with reflected light, is filled with 'green', and the second blank, associated with transmitted light, is filled with 'red'.

This completes the sentence while accurately reflecting the facts and satisfying the student's rhetorical goal.

Key Concept

Synthesizing specific details from notes to achieve a target rhetorical goal.

Estimated Time:1m 0s

Question 118Question →

A robotic arm rotates counterclockwise by $210^\circ$ and then rotates clockwise by $75^\circ$ . What is the net counterclockwise rotation of the robotic arm, in radians?

\frac{3\pi}{4}

\frac{4\pi}{3}

\frac{29\pi}{36}

\frac{3\pi}{5}

Show answer & explanation

Answer: $\frac{3\pi}{4}$

Answer

The correct answer is

\frac{3\pi}{4}

radians.

The correct answer is determined by first finding the net angle of rotation. Subtracting the

75^\circ

clockwise rotation from the

210^\circ

counterclockwise rotation yields a net rotation of

135^\circ

counterclockwise. Converting this angle to radians is done by multiplying

135^\circ

by

\frac{\pi}{180^\circ}

, resulting in

\frac{135\pi}{180}

radians. Dividing both the numerator and denominator by 45 simplifies the fraction to

\frac{3\pi}{4}

radians.

Step-by-Step Solution

1

Calculate the net counterclockwise rotation in degrees.

210^\circ - 75^\circ = 135^\circ

Since counterclockwise rotation is positive and clockwise rotation is negative, we subtract the clockwise rotation from the counterclockwise rotation.

2

Convert the angle from degrees to radians.

135 \times \frac{\pi}{180} = \frac{135\pi}{180}

radians

To convert degrees to radians, multiply the degree measure by the conversion factor

\frac{\pi}{180^\circ}

.

3

Simplify the fraction to its lowest terms.

\frac{3\pi}{4}

radians

Divide both the numerator and the denominator by their greatest common divisor, which is 45.

Key Concept

To convert an angle from degrees to radians, multiply the degree measure by

\frac{\pi}{180^\circ}

and simplify the resulting fraction.

Alternative Method

Alternatively, convert each individual rotation to radians first, then subtract. Convert

210^\circ

to radians:

210 \times \frac{\pi}{180} = \frac{7\pi}{6}

radians. Convert

75^\circ

to radians:

75 \times \frac{\pi}{180} = \frac{5\pi}{12}

radians. Subtract the clockwise rotation from the counterclockwise rotation:

\frac{7\pi}{6} - \frac{5\pi}{12} = \frac{14\pi}{12} - \frac{5\pi}{12} = \frac{9\pi}{12} = \frac{3\pi}{4}

radians.

Estimated Time:1m 30s

Question 119Question →

In the $xy$ -plane, a right triangle has vertices at the origin $O(0, 0)$ , $A(x, 0)$ , and $B(0, y)$ , where $x > 0$ and $y > 0$ . If the length of the hypotenuse $AB$ is $26$ and the slope of the line containing segment $AB$ is $-\frac{12}{5}$ , what is the value of $y$ ?

Show answer & explanation

Answer: 24

Answer

The value of

y

is 24.

The correct answer is 24 because the slope of the line through

A(x, 0)

and

B(0, y)

is

-\frac{y}{x}

. Setting this equal to

-\frac{12}{5}

gives the relationship

y = \frac{12}{5}x

. Substituting this into the Pythagorean theorem equation

x^2 + y^2 = 26^2

yields

x^2 + \left(\frac{12}{5}x\right)^2 = 676

, which simplifies to

\frac{169}{25}x^2 = 676

. Solving for

x

gives

x = 10

, and multiplying by

\frac{12}{5}

gives the value of

y

as 24.

Step-by-Step Solution

1

Express the slope of the line containing segment

AB

in terms of

x

and

y

.

y = \frac{12}{5}x

The line passes through

A(x, 0)

and

B(0, y)

, so its slope is

-\frac{y}{x}

. Setting this equal to the given slope of

-\frac{12}{5}

gives the relationship between

x

and

y

.

2

Apply the Pythagorean theorem to the right triangle

OAB

.

x^2 + y^2 = 676

The lengths of the legs of the right triangle are

x

and

y

, and the hypotenuse is

26

.

3

Substitute the slope relationship into the Pythagorean equation to solve for

x

.

x = 10

Substituting

y = \frac{12}{5}x

gives

x^2 + \frac{144}{25}x^2 = 676

, which simplifies to

\frac{169}{25}x^2 = 676

. Solving for

x

gives

10

.

4

Calculate the value of

y

.

y = 24

Using

x = 10

in

y = \frac{12}{5}x

gives

24

.

Key Concept

Solving right triangle problems in the coordinate plane by combining linear equations (slope) with the Pythagorean theorem.

Question 120Question →

A sign is in the shape of a parallelogram with a base of $18$ inches and a height of $10$ inches. A smaller, similar parallelogram is cut out from the center of the sign. If the scale factor of the smaller parallelogram to the larger parallelogram is $1$ to $3$ , what is the area, in square inches, of the remaining portion of the sign?

Show answer & explanation

Answer: 160

Answer

160

The area of the larger parallelogram is

18 \times 10 = 180

square inches. Since the smaller parallelogram is similar to the larger one with a scale factor of

1

to

3

, the area of the smaller parallelogram is

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

of the area of the larger parallelogram. This gives an area of

180 \times \frac{1}{9} = 20

square inches for the smaller parallelogram. Subtracting this from the larger area yields

180 - 20 = 160

square inches.

Step-by-Step Solution

1

Calculate the area of the larger parallelogram.

A_{\text{larger}} = 180

square inches

The area of a parallelogram is the product of its base and height (

A = bh

).

2

Determine the area of the smaller, similar parallelogram.

A_{\text{smaller}} = 20

square inches

The ratio of the areas of similar figures is the square of the scale factor:

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

.

3

Subtract the smaller area from the larger area to find the remaining area.

A_{\text{remaining}} = 160

square inches

The remaining portion of the sign is the total area minus the cutout area.

Key Concept

The area of similar geometric shapes scales by the square of the linear scale factor.

Answer

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