All practice questions

Question 1561Question →

Although many historians credit Johannes Gutenberg with the invention of movable type in Europe, the technology had already been developed in Asia centuries earlier. In 1041, Chinese alchemist Bi Sheng created the world’s first known movable type system ______ he used individual clay pieces for each character, which could then be rearranged and inked to print multiple pages.

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

, he

: he

; using

he

Show answer & explanation

Answer: : he

Answer

The correct answer is the option containing a colon followed by the pronoun 'he'. A colon is appropriate here because it separates two independent clauses, where the second clause directly explains or expands on the first.

The option containing the colon followed by the pronoun 'he' is correct because the clause before the punctuation mark is independent, and the clause after it is also independent and serves to explain or elaborate on the first clause. In Standard English conventions, a colon is a proper way to link an independent clause to an explanatory clause.

Step-by-Step Solution

1

Analyze the structure of the clauses before and after the blank.

The clause preceding the blank ('In 1041, Chinese alchemist Bi Sheng created the world’s first known movable type system') is independent. The clause following the blank ('he used individual clay pieces...') is also independent and explains the first clause.

Identifying the clause types helps determine which punctuation marks are grammatically permissible to link them.

2

Evaluate the grammatical correctness of each punctuation option.

A colon is correct because the first clause is independent and the second clause explains it. A comma alone creates a comma splice. A semicolon cannot be followed by a dependent participial phrase like 'using'. Leaving no punctuation creates a run-on sentence.

Standard English conventions dictate that a colon must be preceded by an independent clause and can link to an explanatory clause, while commas and semicolons have stricter clause-boundary rules.

Key Concept

Using colons, semicolons, and dashes to link clauses or phrases

Question 1562Question →

Discovered in the late twentieth century near deep-sea hydrothermal vents on the ocean floor, giant tube worms lack a mouth, gut, or digestive tract. Instead, each individual worm relies entirely on millions of symbiotic bacteria housed within the ______ specialized organ, known as the trophosome, which converts hydrogen sulfide into nutrients.

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

worm

worms

worm's

worms'

Show answer & explanation

Answer: worm's

Answer

worm's

The correct answer is the singular possessive form 'worm's'. The sentence refers to the 'specialized organ' belonging to 'each individual worm'. Because 'each individual' is singular, the possessive noun modifying 'specialized organ' must also be singular. Adding an apostrophe followed by 's' to the singular noun 'worm' correctly indicates this singular possessive relationship.

Step-by-Step Solution

1

Analyze the context to determine the relationship between the target noun and the noun phrase that follows it.

The target noun modifies 'specialized organ', indicating ownership or possession of the organ by the worm. Therefore, a possessive form of the noun is required.

Possessive nouns are used to show ownership or association with a following noun.

2

Determine whether the possessive noun should be singular or plural by looking for clues in the sentence.

The phrase 'each individual worm' establishes that the sentence is referring to a single organism. Thus, a singular possessive noun is required.

Singular subjects require singular possessive modifiers to maintain grammatical agreement.

3

Select the correct spelling of the singular possessive form of 'worm'.

The singular possessive form is created by adding an apostrophe and 's' to the end of the singular noun: 'worm's'.

Standard English conventions dictate adding '-\'s' to a singular noun to indicate possession.

Key Concept

Plural and Possessive Nouns

Question 1563Question →

A logistics company uses a delivery drone to transport two types of packages. Let $x$ represent the number of Type A packages and $y$ represent the number of Type B packages in a single flight. The drone can carry at most 12 packages in total. Additionally, to balance the drone, the total weight of the cargo must be at least 15 pounds. Each Type A package weighs 2 pounds, and each Type B package weighs 1 pound. Which of the following combinations of Type A and Type B packages is a viable shipment for the drone?

3 Type A packages and 8 Type B packages

4 Type A packages and 5 Type B packages

5 Type A packages and 6 Type B packages

6 Type A packages and 7 Type B packages

Show answer & explanation

Answer: 5 Type A packages and 6 Type B packages

Answer

5 Type A packages and 6 Type B packages

The correct combination of 5 Type A packages and 6 Type B packages satisfies all system requirements. The total package count of 11 is less than or equal to the drone's limit of 12 packages (

5 + 6 \leq 12

). Furthermore, the total weight of 16 pounds meets the minimum balance requirement of 15 pounds (

2(5) + 6 \geq 15

).

Step-by-Step Solution

1

Write the system of linear inequalities that represents the given constraints.

The capacity constraint is

x + y \leq 12

, and the weight constraint is

2x + y \geq 15

, where

x

and

y

are non-negative integers.

To define the mathematical boundaries for a valid shipment.

2

Substitute the package counts from the correct combination into both inequalities to verify they are satisfied.

For 5 Type A packages (

x = 5

) and 6 Type B packages (

y = 6

):
- Package count:

5 + 6 = 11 \leq 12

(True)
- Cargo weight:

2(5) + 6 = 16 \geq 15

(True)

To confirm that the chosen combination satisfies all system requirements.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 30s

Question 1564Question →

A coffee shop sells two sizes of bagged coffee: a standard bag and a deluxe bag. On Monday, the shop sold $8$ standard bags and $5$ deluxe bags for a total of $\$108$ . On Tuesday, the shop sold $6$ standard bags and $10$ deluxe bags for a total of $\$156$ . What is the price, in dollars, of one deluxe bag of coffee?

Show answer & explanation

Answer: 12

Answer

The price of one deluxe bag of coffee is

12

dollars.

By setting up a system of equations, we represent the total cost on Monday as

8s + 5d = 108

and on Tuesday as

6s + 10d = 156

. Multiplying the first equation by

2

gives

16s + 10d = 216

. Subtracting the second equation yields

10s = 60

, so

s = 6

. Substituting

s = 6

into the first equation yields

8(6) + 5d = 108

, which simplifies to

48 + 5d = 108

, then

5d = 60

, and finally

d = 12

.

Step-by-Step Solution

1

Define the variables and write the system of equations.

Let

s

be the price of a standard bag and

d

be the price of a deluxe bag. The system is:

8s + 5d = 108

6s + 10d = 156

To translate the word problem into mathematical equations.

2

Multiply the first equation by

2

to align the coefficients of

d

.

16s + 10d = 216

To prepare for elimination of the variable

d

.

3

Subtract the second equation from the new equation to solve for

s

.

10s = 60

, which simplifies to

s = 6

.

To eliminate

d

and find the price of a standard bag.

4

Substitute the value of

s

back into one of the original equations to solve for

d

.

8(6) + 5d = 108 \implies 48 + 5d = 108 \implies 5d = 60 \implies d = 12

.

To find the price of a deluxe bag.

Key Concept

Solving systems of linear equations using the elimination method.

Estimated Time:1m 30s

Question 1565Question →

What form of the noun 'machine' correctly completes the blank in the passage?

Fill in the blanks below

In 1843, mathematician Ada Lovelace published extensive notes on Charles Babbage's proposed mechanical computer, which was known as the Analytical Engine. While Babbage focused primarily on calculation, Lovelace realized that the device could do more than just compute numbers; she envisioned the potential to process symbols and music, effectively writing the world's first computer program.

Show answer & explanation

Answer

machine's

The singular possessive noun 'machine's' is correct because the context refers to a single machine (the Analytical Engine) and shows ownership of the noun 'potential' that follows the blank.

Step-by-Step Solution

1

Determine if the noun should be singular or plural based on the context.

The passage refers to a single device (the Analytical Engine / the device), indicating that the singular form of the noun 'machine' is required.

The context refers back to Charles Babbage's proposed mechanical computer, which is singular.

2

Determine if the noun needs to be possessive.

The noun modifies the noun 'potential,' indicating that the potential belongs to the machine. Therefore, the possessive case is required.

A noun modifying another noun to show ownership or association must be in the possessive case.

3

Form the singular possessive spelling of 'machine'.

Add an apostrophe followed by the letter 's' to the singular noun 'machine' to form 'machine's'.

Singular possessive nouns are formed by adding "'s" to the singular noun.

Key Concept

Plural and Possessive Nouns and Pronouns

Estimated Time:1m 0s

Question 1566Question →

An exponential function $g$ is defined by $g(x) = p \cdot q^x$ , where $p$ and $q$ are positive constants. If $g\left(\frac{1}{2}\right) = 6$ and $g\left(\frac{3}{2}\right) = 54$ , what is the value of $g(2)$ ?

Show answer & explanation

Answer: 162

Answer

162

The correct answer is 162. Substituting the coordinates into the function gives the system

p \cdot q^{1/2} = 6

and

p \cdot q^{3/2} = 54

. Dividing the second equation by the first yields

q = 9

. Substituting

q = 9

into the first equation yields

p \cdot 3 = 6

, so

p = 2

. The complete function is

g(x) = 2 \cdot 9^x

, and evaluating

g(2)

gives

2 \cdot 81 = 162

.

Step-by-Step Solution

1

Substitute the given values into the function definition to set up a system of equations.

p \cdot q^{1/2} = 6

and

p \cdot q^{3/2} = 54

To represent the given coordinate relationships using the exponential model parameters.

2

Divide the equation for

g\left(\frac{3}{2}\right)

by the equation for

g\left(\frac{1}{2}\right)

to eliminate

p

and solve for

q

.

q = 9

Using the exponent quotient rule,

\frac{q^{3/2}}{q^{1/2}} = q^{3/2 - 1/2} = q^1 = q

. Dividing the constants gives

\frac{54}{6} = 9

.

3

Substitute

q = 9

back into the first equation and solve for

p

.

p = 2

Evaluating the fractional exponent gives

9^{1/2} = \sqrt{9} = 3

. Substituting this in yields

3p = 6

, which simplifies to

p = 2

.

4

Evaluate the completed exponential function

g(x) = 2 \cdot 9^x

at

x = 2

.

162

Calculating

g(2) = 2 \cdot 9^2 = 2 \cdot 81 = 162

solves for the target value.

Key Concept

Solving a system of exponential equations using fractional exponent rules and base evaluation.

Question 1567Question →

A geophysicist models the temperature, $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of a rock layer during a deep-crust drilling project using a linear function of the depth, $d$ , in kilometers ( $\text{km}$ ), below the surface. According to the model, for every increase in depth of $0.8\text{ km}$ , the temperature of the rock increases by $22^\circ\text{C}$ . At a depth of $2.4\text{ km}$ , the temperature of the rock is $81^\circ\text{C}$ . According to the model, at what depth, in kilometers, will the temperature of the rock be $114^\circ\text{C}$ ?

Show answer & explanation

Answer: 3.6

Answer

The temperature of the rock will reach

114^\circ\text{C}

at a depth of

3.6

kilometers.

The correct answer is

3.6

. The rate of change of temperature with depth is

\frac{22^\circ\text{C}}{0.8\text{ km}} = 27.5^\circ\text{C/km}

. The linear relationship between temperature

T

and depth

d

can be modeled by

T = 27.5d + T_0

, where

T_0

is the temperature at the surface. Substituting the known values

d = 2.4

and

T = 81

into the model gives

81 = 27.5(2.4) + T_0

, which simplifies to

81 = 66 + T_0

. Solving for

T_0

yields

T_0 = 15

. Thus, the model is

T = 27.5d + 15

. To find the depth when the temperature is

114^\circ\text{C}

, substitute

T = 114

into the model:

114 = 27.5d + 15

. Subtracting

15

from both sides gives

99 = 27.5d

, and dividing by

27.5

yields

d = 3.6

.

Step-by-Step Solution

1

Find the rate of change (slope) of the temperature with respect to depth.

The slope is

27.5^\circ\text{C/km}

.

The temperature increases by

22^\circ\text{C}

for every

0.8\text{ km}

of depth, so the rate of change is

\frac{22^\circ\text{C}}{0.8\text{ km}} = 27.5^\circ\text{C/km}

.

2

Set up a linear model and find the surface temperature (y-intercept).

The linear model is

T = 27.5d + 15

.

Using the slope-intercept form

T = md + T_0

and substituting the given values

d = 2.4

and

T = 81

gives

81 = 27.5(2.4) + T_0

, which simplifies to

81 = 66 + T_0

, so

T_0 = 15

.

3

Substitute the target temperature into the linear model and solve for the target depth.

d = 3.6

Substitute

T = 114

into the equation

T = 27.5d + 15

to get

114 = 27.5d + 15

. Subtracting

15

from both sides gives

99 = 27.5d

. Dividing both sides by

27.5

yields

d = 3.6

.

Key Concept

Interpreting slope and solving for values in a linear relationship context.

Alternative Method

Find the required temperature increase:

114^\circ\text{C} - 81^\circ\text{C} = 33^\circ\text{C}

. Set up a proportion using the rate of

22^\circ\text{C}

increase per

0.8\text{ km}

to find the change in depth

\Delta d

:

\frac{33^\circ\text{C}}{\Delta d} = \frac{22^\circ\text{C}}{0.8\text{ km}} \implies \Delta d = \frac{33 \times 0.8}{22} = 1.2\text{ km}

. Add this change in depth to the initial depth to find the final depth:

2.4 + 1.2 = 3.6\text{ km}

.

Estimated Time:2m 0s

Question 1568Question →

A radioactive substance decays according to the equation $N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{8}}$ , where $N(t)$ is the amount of the substance remaining after $t$ days, and $N_0$ is the initial amount. If the substance decays to $\frac{1}{16}$ of its initial amount in $d$ days, what is the value of $d$ ?

32

4

2

128

Show answer & explanation

Answer: 32

Answer

32

The correct answer is 32. The remaining amount of the substance is given as

\frac{1}{16}

of the initial amount, so

\frac{N(d)}{N_0} = \frac{1}{16}

. Substituting

d

for

t

in the decay formula yields

\left(\frac{1}{2}\right)^{\frac{d}{8}} = \frac{1}{16}

. Since

\frac{1}{16} = \left(\frac{1}{2}\right)^4

, we can set the exponents equal to each other:

\frac{d}{8} = 4

. Solving for

d

gives

d = 32

.

Step-by-Step Solution

1

Set up the equation for the remaining fraction of the substance.

\frac{N(d)}{N_0} = \frac{1}{16}

The problem states the substance decays to

\frac{1}{16}

of its initial amount

N_0

in

d

days.

2

Substitute the given decay formula into the equation.

\left(\frac{1}{2}\right)^{\frac{d}{8}} = \frac{1}{16}

By replacing

t

with

d

in the formula

\frac{N(t)}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{8}}

, we get the expression for the remaining fraction.

3

Rewrite

\frac{1}{16}

as a power with base

\frac{1}{2}

to solve for the exponent.

\left(\frac{1}{2}\right)^4 = \frac{1}{16}

Since

2^4 = 16

, the fraction

\frac{1}{16}

is equivalent to

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

.

4

Equate the exponents and solve for

d

.

\frac{d}{8} = 4 \implies d = 32

Since the bases are equal, their exponents must be equal. Multiplying both sides by 8 isolates

d

.

Key Concept

Solving exponential equations by equating bases

Question 1569Question →

In the system of equations below, $x$ and $y$ are real numbers.

4^x \cdot 8^y = 128

9^x \cdot 3^y = 243

What is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is

2

.

To solve the system of equations, express all terms in each equation using common bases. In the first equation,

4^x \cdot 8^y = 128

can be written as

(2^2)^x \cdot (2^3)^y = 2^7

, which simplifies to

2^{2x+3y} = 2^7

, meaning

2x + 3y = 7

. In the second equation,

9^x \cdot 3^y = 243

can be written as

(3^2)^x \cdot 3^y = 3^5

, which simplifies to

3^{2x+y} = 3^5

, meaning

2x + y = 5

. Subtracting the two equations gives

(2x + 3y) - (2x + y) = 7 - 5 \implies 2y = 2 \implies y = 1

. Substituting

y = 1

back into

2x + y = 5

gives

2x + 1 = 5 \implies 2x = 4 \implies x = 2

.

Step-by-Step Solution

1

Rewrite the first equation using a common base of 2.

2x + 3y = 7

By converting

4^x

to

(2^2)^x = 2^{2x}

,

8^y

to

(2^3)^y = 2^{3y}

, and

128

to

2^7

, we can equate the exponents:

2x + 3y = 7

.

2

Rewrite the second equation using a common base of 3.

2x + y = 5

By converting

9^x

to

(3^2)^x = 3^{2x}

, keeping

3^y

, and converting

243

to

3^5

, we can equate the exponents:

2x + y = 5

.

3

Solve the system of linear equations for

y

.

y = 1

Subtracting

2x + y = 5

from

2x + 3y = 7

eliminates

x

, leaving

2y = 2

, which gives

y = 1

.

4

Substitute

y = 1

into one of the linear equations to solve for

x

.

x = 2

Substituting

y = 1

into

2x + y = 5

yields

2x + 1 = 5

, which simplifies to

2x = 4

, so

x = 2

.

Key Concept

Solving systems of exponential equations by converting to a common base and applying exponent laws.

Question 1570Question →

Which of the following is equivalent to the expression $\frac{3x^2 - 5x - 4}{x - 2}$ for all $x \neq 2$ ?

3x - 1 - \frac{6}{x - 2}

3x + 1 - \frac{2}{x - 2}

3x + 1 + \frac{2}{x - 2}

3x - 5 - \frac{4}{x - 2}

Show answer & explanation

Answer: $3x + 1 - \frac{2}{x - 2}$

Answer

3x + 1 - \frac{2}{x - 2}

The correct answer represents the equivalent expression obtained by performing polynomial division on the rational expression. Dividing the numerator

3x^2 - 5x - 4

by the denominator

x - 2

yields a quotient of

3x + 1

and a remainder of

-2

. This can be written in the form of the quotient plus the remainder over the divisor, resulting in

3x + 1 - \frac{2}{x - 2}

.

Step-by-Step Solution

1

Divide the leading term of the numerator,

3x^2

, by the leading term of the denominator,

x

.

The first term of the quotient is

3x

.

This starts the polynomial long division process.

2

Multiply

3x

by the divisor

(x - 2)

and subtract the result from the numerator.

(3x^2 - 5x - 4) - (3x^2 - 6x) = x - 4

.

Subtracting the multiplied term helps find the remainder of the first division step.

3

Divide the leading term of the remaining expression,

x

, by the leading term of the divisor,

x

.

The second term of the quotient is

1

.

To continue the division process with the remaining terms.

4

Multiply

1

by the divisor

(x - 2)

and subtract the result from

x - 4

.

(x - 4) - (x - 2) = -2

.

This step determines the final remainder of

-2

because the degree of the remainder is now less than the degree of the divisor.

5

Express the final result as the sum of the quotient and the remainder divided by the divisor.

3x + 1 - \frac{2}{x - 2}

To construct the equivalent algebraic expression.

Key Concept

Equivalent Algebraic Expressions

Question 1571Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + bx + c$ has a vertex at $(h, k)$ . The graph of $f$ is translated $4$ units to the right and $5$ units up to form the graph of a quadratic function $g$ . If the graph of $g$ passes through the point $(1, 16)$ and has a $y$ -intercept of $13$ , what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

The value of

k

is

12

.

By writing the function in vertex form as

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

, we can apply the transformations directly. Translating the graph

4

units to the right and

5

units up gives the function

g(x) = f(x-4) + 5 = -(x-h-4)^2 + k+5

. Using the given points

g(0) = 13

and

g(1) = 16

, we set up a system of equations:

-(h+4)^2 + k+5 = 13

and

-(h+3)^2 + k+5 = 16

. Solving these simultaneously gives

h = -2

and

k = 12

.

Step-by-Step Solution

1

Write the function

f(x)

in vertex form.

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

Since the leading coefficient of

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

is

-1

, the vertex form must have a coefficient of

-1

.

2

Represent the translated function

g(x)

.

g(x) = -(x-h-4)^2 + k+5

Translating a function

f(x)

by

4

units to the right and

5

units up yields

g(x) = f(x-4) + 5

.

3

Use the

y

-intercept of

g(x)

to write the first equation.

k - (h+4)^2 = 8

The

y

-intercept is the value of the function at

x = 0

, so

g(0) = 13

.

4

Use the point

(1, 16)

on

g(x)

to write the second equation.

k - (h+3)^2 = 11

The graph passing through

(1, 16)

means

g(1) = 16

.

5

Solve the system of equations for

h

and

k

.

h = -2

and

k = 12

Equating the expressions for

k

gives

(h+4)^2 + 8 = (h+3)^2 + 11

, which simplifies to

2h = -4

, so

h = -2

. Substituting

h = -2

into

k = (h+4)^2 + 8

yields

k = 12

.

Key Concept

Translating quadratic functions and solving systems of quadratic equations.

Question 1572Question →

In the inequality $3(4 - 2x) - 5(x - c) \geq -7$ , where $c$ is a constant, the solution set consists of all values of $x$ such that $x \leq 3$ . What is the value of $c$ ?

Show answer & explanation

Answer: 2.8

Answer

2.8

To find the value of

c

, first simplify the inequality

3(4 - 2x) - 5(x - c) \geq -7

by expanding the terms using the distributive property, which yields

12 - 6x - 5x + 5c \geq -7

. Combining like terms gives

-11x + 12 + 5c \geq -7

. Next, isolate the variable term by subtracting

12

and

5c

from both sides to obtain

-11x \geq -19 - 5c

. Dividing both sides of the inequality by

-11

and reversing the inequality sign results in

x \leq \frac{19 + 5c}{11}

. Given that the solution set consists of all values of

x

such that

x \leq 3

, the boundary value

\frac{19 + 5c}{11}

must equal

3

. Solving the equation

\frac{19 + 5c}{11} = 3

gives

19 + 5c = 33

, which simplifies to

5c = 14

and results in

c = 2.8

.

Step-by-Step Solution

1

Distribute the constants on the left side of the inequality.

12 - 6x - 5x + 5c \geq -7

Applying the distributive property simplifies the parentheses.

2

Combine the variable terms.

-11x + 12 + 5c \geq -7

Grouping like terms simplifies the inequality.

3

Subtract

12

and

5c

from both sides of the inequality.

-11x \geq -19 - 5c

This isolates the term containing

x

.

4

Divide both sides by

-11

and reverse the direction of the inequality.

x \leq \frac{19 + 5c}{11}

Dividing by a negative number requires flipping the inequality sign to maintain equivalence.

5

Set the algebraic boundary

\frac{19 + 5c}{11}

equal to the given boundary value of

3

and solve for

c

.

c = 2.8

Since the solution set is

x \leq 3

, the boundary values must be equivalent.

Key Concept

Solving multi-step linear inequalities in one variable containing parameters.

Question 1573Question →

The graph of the equation $ax + by = 24$ , where $a$ and $b$ are constants, is a line in the $xy$ -plane. If this line passes through the points $(2, 9)$ and $(6, 3)$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is

5

.

Substituting the coordinates of the two points

(2, 9)

and

(6, 3)

into the given linear equation

ax + by = 24

yields a system of two equations:

2a + 9b = 24

and

6a + 3b = 24

. Simplifying the second equation gives

2a + b = 8

, which means

b = 8 - 2a

. Substituting this expression for

b

into the first equation yields

2a + 9(8 - 2a) = 24

, which simplifies to

-16a = -48

, or

a = 3

. Plugging

a = 3

back into

b = 8 - 2a

gives

b = 2

. Thus, the value of

a + b

is

3 + 2 = 5

.

Step-by-Step Solution

1

Substitute the point

(2, 9)

into the equation

ax + by = 24

.

2a + 9b = 24

Since the line passes through the point

(2, 9)

, the coordinates must satisfy the equation of the line.

2

Substitute the point

(6, 3)

into the equation

ax + by = 24

.

6a + 3b = 24

Since the line passes through the point

(6, 3)

, the coordinates must satisfy the equation of the line.

3

Solve the system of equations for

a

and

b

.

a = 3

and

b = 2

To find the values of the constants

a

and

b

, we solve the linear system: (1)

2a + 9b = 24

and (2)

6a + 3b = 24

. Dividing the second equation by 3 gives

2a + b = 8

, or

b = 8 - 2a

. Substituting this into the first equation gives

2a + 9(8 - 2a) = 24 \implies 2a + 72 - 18a = 24 \implies -16a = -48 \implies a = 3

. Then,

b = 8 - 2(3) = 2

.

4

Add the values of

a

and

b

.

a + b = 5

The question asks for the value of

a + b

.

Key Concept

Solving systems of linear equations derived from coordinate substitution in a two-variable linear equation.

Estimated Time:1m 30s

Question 1574Question →

For a constant $k$ , the inequality $2(3 - x) - k(x + 5) > 4$ has the solution set $x < -3$ . What is the value of $k$ ?

-1

2

4

-0.5

Show answer & explanation

Answer: $4$

Answer

4

The correct answer is

4

. Expanding the inequality

2(3 - x) - k(x + 5) > 4

gives

6 - 2x - kx - 5k > 4

. Grouping the

x

terms yields

(-2 - k)x > 5k - 2

. Because the given solution set is

x < -3

, the inequality sign must flip, indicating that the coefficient

-2 - k

is negative. Dividing by this negative coefficient yields

x < \frac{5k - 2}{-2 - k}

. Setting the boundary expression equal to the boundary of the solution set gives

\frac{5k - 2}{-2 - k} = -3

. Multiplying both sides by

-2 - k

results in

5k - 2 = 6 + 3k

. Subtracting

3k

and adding

2

to both sides gives

2k = 8

, which simplifies to

k = 4

.

Step-by-Step Solution

1

Distribute the constants in the inequality

6 - 2x - kx - 5k > 4

To clear parentheses and prepare to group terms.

2

Group the

x

terms and constant terms

(-2 - k)x > 5k - 2

To isolate the variable

x

on one side of the inequality.

3

Divide by the coefficient of

x

and flip the inequality direction

x < \frac{5k - 2}{-2 - k}

Since the solution set is

x < -3

, the direction of the inequality must flip from

>

to

<

, meaning the coefficient

-2 - k

must be negative.

4

Set the boundary value equal to

-3

and solve for

k

k = 4

The boundary of the solution set must be equal to

-3

. Solving

\frac{5k - 2}{-2 - k} = -3

yields

5k - 2 = 6 + 3k

, which simplifies to

2k = 8

, so

k = 4

.

Key Concept

Solving linear inequalities in one variable with symbolic coefficients, accounting for direction flips when dividing by negative quantities.

Question 1575Question →

\begin{aligned} (x - 4)^2 + (y + 1)^2 &= 18 \\ x + y &= k \end{aligned}

In the system of equations above, $k$ is a constant. If the system has exactly one real solution, what is the sum of all possible values of $k$ ?

-6

2

6

10

Show answer & explanation

Answer: 6

Answer

The sum of all possible values of

k

is

6

.

The system of equations has exactly one real solution when the line is tangent to the circle. The circle

(x - 4)^2 + (y + 1)^2 = 18

has center

(4, -1)

and radius

\sqrt{18} = 3\sqrt{2}

. The distance from the center of the circle to the line

x + y - k = 0

is given by

\frac{|4 - 1 - k|}{\sqrt{1^2 + 1^2}} = \frac{|3 - k|}{\sqrt{2}}

. Setting this distance equal to the radius yields

\frac{|3 - k|}{\sqrt{2}} = 3\sqrt{2} \implies |3 - k| = 6

. Solving this equation gives

k = 9

and

k = -3

. The sum of these possible values is

9 + (-3) = 6

.

Step-by-Step Solution

1

Relate the algebraic condition of having exactly one solution to the geometric relationship between the circle and the line.

The line must be tangent to the circle, meaning the perpendicular distance from the center of the circle to the line must equal the radius of the circle.

A system consisting of a circle and a line has exactly one real solution if and only if the line is tangent to the circle.

2

Identify the center and radius of the circle from the equation

(x - 4)^2 + (y + 1)^2 = 18

.

The center of the circle is

(4, -1)

and the radius is

r = \sqrt{18} = 3\sqrt{2}

.

The standard form of a circle equation is

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

, where

(h, k)

is the center and

r

is the radius.

3

Express the line equation in standard form

Ax + By + C = 0

and set up the perpendicular distance formula.

The line equation is

x + y - k = 0

. The distance from the center

(4, -1)

to the line is

d = \frac{|(1)(4) + (1)(-1) - k|}{\sqrt{1^2 + 1^2}} = \frac{|3 - k|}{\sqrt{2}}

.

The perpendicular distance from a point

(x_0, y_0)

to a line

Ax + By + C = 0

is given by

d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}

.

4

Set the distance equal to the radius of the circle and solve the absolute value equation for

k

.

\frac{|3 - k|}{\sqrt{2}} = 3\sqrt{2} \implies |3 - k| = 6

. This yields two equations:

3 - k = 6 \implies k = -3

, and

3 - k = -6 \implies k = 9$.

Setting the distance equal to the radius ensures that the line is tangent to the circle, resulting in exactly one point of intersection.

5

Calculate the sum of all possible values of

k

.

Sum =

-3 + 9 = 6

.

The question asks for the sum of all values of

k

that satisfy the tangency condition.

Key Concept

Determining the relationship between a line and a circle in a system of equations by setting the perpendicular distance from the center of the circle equal to its radius.

Alternative Method

Substitute

y = k - x

into the circle equation to obtain a quadratic equation in

x

:

(x - 4)^2 + (k - x + 1)^2 = 18

. Expanding and combining like terms yields

2x^2 - 2(k + 5)x + (k^2 + 2k - 1) = 0

. For the system to have exactly one real solution, the discriminant

\Delta

of this quadratic equation must be zero:

\Delta = [-2(k + 5)]^2 - 4(2)(k^2 + 2k - 1) = 0

. Simplifying this equation leads to

4(k^2 + 10k + 25) - 8(k^2 + 2k - 1) = 0 \implies k^2 - 6k - 27 = 0

. By Vieta's formulas, the sum of the solutions to this quadratic equation for

k

is given by

-\frac{b}{a} = -\frac{-6}{1} = 6

.

Estimated Time:2m 30s

Question 1576Question →

A system of equations consists of the quadratic function $f(x) = -x^2 + 6x - 2$ and the linear function $g(x) = 2x + 1$ . If the graphs of $f$ and $g$ intersect at the points $(x_1, y_1)$ and $(x_2, y_2)$ in the $xy$ -plane, what is the value of $y_1 + y_2$ ?

Show answer & explanation

Answer: 10

Answer

10

By setting the two equations equal to each other, we obtain

-x^2 + 6x - 2 = 2x + 1

. Rearranging terms to one side gives

x^2 - 4x + 3 = 0

, which factors as

(x-1)(x-3) = 0

. The x-coordinates of the intersection points are

x = 1

and

x = 3

. Substituting these into the linear equation gives the y-coordinates: for

x = 1

,

y = 2(1) + 1 = 3

; for

x = 3

,

y = 2(3) + 1 = 7

. Summing these y-coordinates gives

3 + 7 = 10

.

Step-by-Step Solution

1

Set the quadratic function equal to the linear function to find the x-coordinates of the intersection points.

-x^2 + 6x - 2 = 2x + 1

At the points of intersection, the y-values of both functions must be equal.

2

Rearrange the equation into standard quadratic form and solve for

x

by factoring.

x^2 - 4x + 3 = 0 \Rightarrow (x - 1)(x - 3) = 0 \Rightarrow x = 1 \text{ or } x = 3

Factoring the quadratic equation gives the x-coordinates of the intersection points.

3

Substitute the x-coordinates back into the linear equation to find the corresponding y-coordinates.

For

x_1 = 1

,

y_1 = 2(1) + 1 = 3

. For

x_2 = 3

,

y_2 = 2(3) + 1 = 7

.

The linear equation

g(x) = 2x + 1

is simpler to use for finding the y-coordinates than the quadratic equation.

4

Sum the two y-coordinates.

y_1 + y_2 = 3 + 7 = 10

The question asks for the sum of the y-coordinates of the intersection points.

Key Concept

Solving a system consisting of a linear equation and a quadratic equation by substitution or equating them.

Question 1577Question →

Based on the grammar and context of the passage, what form of the noun 'dragonfly' correctly completes the blank?

Fill in the blanks below

Unlike most insects, which rely on simple muscle contractions to flap their wings, a dragonfly utilizes direct muscle attachments to control each wing independently. This unique anatomical feature allows the maneuverability to surpass that of almost any other aerial predator, enabling it to hover, fly backward, and change direction mid-flight with remarkable precision.

Show answer & explanation

Answer

dragonfly's

The singular possessive noun 'dragonfly's' correctly attributes the maneuverability to a single dragonfly. The singular context is established by the phrase 'a dragonfly' in the first sentence and the pronoun 'it' in the second sentence. To form a singular possessive, an apostrophe and an 's' must be added to the singular noun.

Step-by-Step Solution

1

Analyze the context to determine if the noun should be singular or plural.

The passage introduces the subject in the singular form ('a dragonfly') and subsequently uses the singular pronoun 'it' ('enabling it to hover'). Therefore, the target noun must be singular.

Identifying the number of the noun ensures correct agreement with the surrounding context.

2

Determine if the noun requires a possessive form.

The noun modifies the resource 'maneuverability', indicating that the maneuverability belongs to the dragonfly. Thus, a possessive case is required.

Identifying the case of the noun determines the correct punctuation structure.

3

Form the singular possessive noun.

Adding an apostrophe followed by 's' to the singular noun 'dragonfly' yields 'dragonfly's'.

Applying standard English orthography rules for singular possessives provides the correct spelling.

Key Concept

Plural and Possessive Nouns

Estimated Time:1m 0s

Question 1578Question →

The quadratic function $f$ is defined by $f(x) = 3(x - 2)^2 + 5$ . In the $xy$ -plane, the graph of $y = f(x)$ intersects the $y$ -axis at the point $(0, c)$ . What is the value of $c$ ?

Show answer & explanation

Answer: 17

Answer

The correct answer is 17.

To find the

y

-intercept of the graph of

y = f(x)

, we substitute

x = 0

into the equation:

f(0) = 3(0 - 2)^2 + 5

. Simplifying the expression inside the parentheses gives

3(-2)^2 + 5

. Squaring

-2

gives

4

, making the expression

3(4) + 5 = 12 + 5 = 17

. Therefore, the graph intersects the

y

-axis at the point

(0, 17)

, which means the value of

c

is 17.

Step-by-Step Solution

1

Set

x = 0

to find the point where the graph intersects the

y

-axis.

f(0) = 3(0 - 2)^2 + 5

The

y

-intercept of any graph in the

xy

-plane occurs when the

x

-coordinate is equal to

0

.

2

Perform the arithmetic evaluation to find the value of

f(0)

.

f(0) = 17

First, evaluate the term inside the parentheses:

0 - 2 = -2

. Squaring

-2

gives

4

. Then, multiply

3

by

4

to get

12

. Finally, add

5

to obtain

17

.

Key Concept

Evaluating a quadratic function at

x = 0

to determine its

y

-intercept.

Question 1579Question →

In the quadratic equation $x^2 + bx + c = 0$ , $b$ and $c$ are integers such that $b + c = 10$ . If the equation has no real solutions, what is the maximum possible value of $c$ ?

18

19

14

6

Show answer & explanation

Answer: 18

Answer

The maximum possible value of

c

is 18.

For the quadratic equation to have no real solutions, the discriminant

b^2 - 4ac

must be negative. Given

a=1

, this simplifies to

b^2 - 4c < 0

. Using the constraint

b + c = 10

, we can express

c

as

10 - b

. Substituting this into the inequality gives

b^2 - 4(10 - b) < 0

, which simplifies to

b^2 + 4b - 40 < 0

. Solving the boundary equation

b^2 + 4b - 40 = 0

using the quadratic formula gives

b = -2 \pm \sqrt{44}

. Since

\sqrt{44} \approx 6.63

, the boundary values of

b

are approximately

-8.63

and

4.63

. Thus, the inequality holds for

-8.63 < b < 4.63

. To maximize

c = 10 - b

, we need the smallest integer value for

b

in this range, which is

b = -8

. Substituting

b = -8

gives the maximum value of

c = 10 - (-8) = 18

.

Step-by-Step Solution

1

State the condition for the quadratic equation to have no real solutions.

The discriminant must be negative:

b^2 - 4ac < 0

. Since

a = 1

, this simplifies to

b^2 - 4c < 0

.

A quadratic equation has no real solutions if and only if its discriminant is strictly less than zero.

2

Express

c

in terms of

b

using the given equation

b + c = 10

, and substitute it into the discriminant inequality.

Since

c = 10 - b

, substituting gives

b^2 - 4(10 - b) < 0

, which simplifies to

b^2 + 4b - 40 < 0

.

Substituting one variable in terms of the other allows us to solve a single-variable quadratic inequality.

3

Find the boundary values of

b

by solving the equation

b^2 + 4b - 40 = 0

using the quadratic formula.

b = \frac{-4 \pm \sqrt{4^2 - 4(1)(-40)}}{2} = \frac{-4 \pm \sqrt{176}}{2} = -2 \pm \sqrt{44}

. Since

\sqrt{44} \approx 6.63

, the boundaries are approximately

b \approx -8.63

and

b \approx 4.63

.

The roots of the quadratic equation define the boundaries of the interval that satisfies the inequality.

4

Determine the interval for

b

that satisfies the inequality and identify the value of

b

that maximizes

c

.

The inequality

b^2 + 4b - 40 < 0

is satisfied when

-8.63 < b < 4.63

. Since

c = 10 - b

, maximizing

c

requires selecting the smallest possible integer value of

b

in this range, which is

b = -8

.

Subtracting a smaller number from 10 results in a larger value for

c

.

5

Calculate the maximum value of

c

using the optimal value of

b

.

c = 10 - (-8) = 18

.

Substituting the minimum integer value of

b

gives the maximum integer value of

c

.

Key Concept

Analyzing the discriminant of a quadratic equation to determine the nature of its roots and using substitution to solve optimization problems under constraints.

Question 1580Question →

If $z$ is a positive number such that $z^2 - 4z - 12 = 0$ , what is the value of $z$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

To find the positive value of

z

that satisfies the equation

z^2 - 4z - 12 = 0

, we factor the quadratic expression. Finding two numbers that multiply to

-12

and add to

-4

gives

-6

and

2

. Thus, the equation can be factored as

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

. Setting each factor to zero yields the solutions

z = 6

and

z = -2

. Since the problem states that

z

is a positive number, we discard the negative solution, leaving

6

as the final answer.

Step-by-Step Solution

1

Factor the quadratic equation

z^2 - 4z - 12 = 0

.

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

To solve the quadratic equation by finding two numbers that multiply to -12 and add to -4.

2

Solve for the roots of the equation.

z = 6

or

z = -2

Set each factor equal to zero and solve for

z

.

3

Apply the constraint that

z

must be positive.

z = 6

Discard the negative solution

z = -2

as the problem specifies

z

must be a positive number.

Key Concept

Solving quadratic equations by factoring

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Alternative Method

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